**Stuart Richer, Jane Cho, William Stiles, Marc Levin, James S. Wrobel, Michael Sinai and Carla Thomas**

**Abstract:** Purpose: A challenge in ocular preventive medicine is identification of patients with early pathological retinal damage that might benefit from nutritional intervention. The purpose of this study is to evaluate retinal thinning (RT) in early atrophic age-related macular degeneration (AMD) against visual function data from the Zeaxanthin and Visual Function (ZVF) randomized double masked placebo controlled clinical trial (FDA IND #78973). Methods: Retrospective, observational case series of medical center veterans with minimal visible AMD retinopathy (AREDS Report #18 simplified grading 1.4/4.0 bilateral retinopathy). Foveal and extra-foveal four quadrant SDOCT RT measurements were evaluated in *n* = 54 clinical and ZVF AMD patients. RT by age was determined and compared to the OptoVue SD OCT normative database. RT by quadrant in a subset of *n* = 29 ZVF patients was correlated with contrast sensitivity and parafoveal blue cone increment thresholds. Results: Foveal RT in AMD patients and non-AMD patients was preserved with age. Extrafoveal regions, however, showed significant slope differences between AMD patients and non-AMD patients, with the superior and nasal quadrants most vulnerable to retinal thinning (sup quad: í5.5 ȝm/decade thinning *vs.* Non-AMD: í1.1 ȝm/decade, *P* < 0.02; nasal quad: í5.0 ȝm/decade thinning *vs.* Non-AMD: í1.0 ȝm/decade, *P* < 0.04). Two measures of extrafoveal visual deterioration were correlated: A significant inverse correlation between % RT and contrast sensitivity (*r* = í0.33, *P* = 0.01, 2 Tailed Paired T) and an elevated extrafoveal increment blue cone threshold (*r* = +0.34, *P* = 0.01, 2 Tailed T). Additional SD OCT RT data for the non-AMD oldest age group (ages 82–91) is needed to fully substantiate the model. Conclusion: A simple new SD OCT clinical metric called "% extra-foveal RT" correlates well with functional visual loss in early AMD patients having minimal visible retinopathy. This metric can be used to follow the effect of repleting ocular nutrients, such as zinc, antioxidants, carotenoids, *n*-3 essential fats, resveratrol and vitamin D.

Reprinted from *Nutrients*. Cite as: Richer, S.; Cho, J.; Stiles, W.; Levin, M.; Wrobel, J.S.; Sinai, M.; Thomas, C. Retinal Spectral Domain Optical Coherence Tomography in Early Atrophic Age-Related Macular Degeneration (AMD) and a New Metric for Objective Evaluation of the Efficacy of Ocular Nutrition. *Nutrients* **2012**, *4*, 1812-1827.

#### **1. Introduction**

Age-related macular degeneration (AMD) is the leading cause of vision loss in both developed and developing countries [1–3]. The initial clinical stereoscopic ophthalmoscopic manifestations of the dry form of the disease are drusen formation and RPE hypo/hyper pigmentation. These signs have been used in large scale "ocular nutrient" studies such as AREDS I and II to monitor disease and stratify risk of progression. Yet much pathology is hidden below the surface of the retina. Classic fundus observation and photography even with stereoscopic enhancement by a highly trained physician often reveals merely the gross manifestations of moderate and advanced disease. Emerging sensitive, new technologies include retinal pigment epithelium autofluorescence imaging [4], multi spectral retinal imaging (*i.e*., Annidis Health Systems, Ottawa, Canada) [5], and high resolution spectral domain OCT imaging for more subtle changes of the retina and choroid [6]. Beyond imaging, lipofuscin is the most consistent and phylogenically constant marker of cellular aging that can be useful for managing AMD [7]. Furthermore, autofluorescence of the A2E fluorophore within the RPE is utilized within AREDS 2 to evaluate geographic atrophic edge expansion [8]. These new technologies, when broadly evaluated in conjunction with preventive medicine/nutritional intervention, may assist eye physicians in preventing the development of moderate and advanced catastrophic AMD.

The Optovue® RTVue (Fremont, CA) was the first commercially available spectral/Fourier domain optical coherence tomography (SD OCT) instrument in the United States. This SD OCT utilizes a low-coherence super luminescent diode light source at a near infrared wavelength of 840 nm (bandwidth 50 nm), and is capable of achieving 5 m axial tissue resolution. The MM5 scan pattern provides full retinal thickness measurements in the macula as well as nine summary parameters from a 9-zone ETDRS grid.

Clinical observations stimulated the design of this study. We observed that patients with very early, and often subclinical atrophic AMD, display various degrees of parafoveal thinning using the SD OCT MM5 protocol. For example, Figure 1a depicts the right eye MM5 and contrast sensitivity function (CSF) of four patients with early atrophic AMD, mild and similar lens opacification rating using the LOCS III grading system [9] and 20/20 or better Snellen acuity. The green areas on the thickness maps show regions that have within normal thickness values (compared to the RTVue normative database). The blue areas show regions that have significant thinning (*P* < 0.05). Note that with increasing parafoveal geographic thinning, there is a dramatic declining contrast sensitivity function (CSF) "Area under the curve" (AUC) value. None of these four representative eyes had additional pathology that would decrease the CSF function, *i.e.*, uncorrected refractive error, clinically significant cataract, diabetes, glaucoma or neurodegenerative disease. Figure 1b depicts an image simulation of a high-risk driver-pedestrian traffic scene generated when each of these four CSF functions is analyzed with image processing software (Functional Vision Analyzer®, Stereo Optical, Inc., Chicago, IL). The last driver in particular, with 92% parafoveal + perifoveal thinning preserves his central foveal thickness, has 20/20 Snellen acuity yet profound CSF AUC loss, profound distance visual disability and profound simulated road-scene image degradation. This clinical observation led us to the hypothesis that retinal thinning is quickly measured by SD OCT, may be correlated with subclinical visual disability, and may therefore serve as a new efficient objective clinical metric for diagnosing early atrophic AMD as well as serving as a new adjunct measure of progression.

**Figure 1.** Retinal thinning, declining contrast sensitivity and simulated impaired driver visibility. (**a**) Spectral Domain OCT MM5 plots of early atrophic AMD phakic retinas from four eyes of four different patients (subject Z22 9% extra-foveal thinning (EFT); Z23 28% EFT; Z18 58% EFT and Z52 92% EFT) in the Zeaxanthin and Vision Function Study (ZVF) FDA IND #78973—see [10]. These subjects all had similar lens opacification by LOCS III analysis and 20/20 or better ETDRS visual acuity. Adjacent CSF plots indicate declining contrast sensitivity with increasing loss of parafoveal retinal thickness. (**b**) Image simulation of a traffic scene of a little girl next to a car computed from each CSF function shown to the left. Images correlate with the monocular CSF of these four patients (clockwise upper left: Z22, Z23, Z18 and Z52), using image processing software (Functional Vision Analyzer®) donated by Stereo Optical, Inc., Chicago, IL.

#### **2. Methods**

Subjects were included from two data sets: a subset of the Zeaxanthin and Vision Function (ZVF) study data set and an additional de-identified clinical OCT data acquired within the Eye Clinic, DVA Medical Center, North Chicago, IL, during routine clinical care. ZVF is a prospective 12 month *n* = 119 eye, randomized, placebo controlled (RCT) trial evaluating one degree macular pigment optical density (MPOD) and distribution (MP), visual acuity, CSF, glare recovery, shape discrimination, large field B/Y color vision threshold, and lipofuscin pattern distributions following supplementation with the carotenoids: zeaxanthin at 8 mg, lutein at 9 mg or a combination of both. ZVF was approved by the R&D and Human Subjects Committees of the Department of Veterans' Affairs Medical Center, Hines, IL under the auspices of the FDA (www.ClinicalTrials.Gov IND#78.973; Richer *et al.* [10]). A total of 56 eyes of twenty nine subjects (27 males and 2 females) had completed ZVF with available post-research study SD OCT scans. In the case of the second larger observational SD OCT data set, de-identified clinical data on a random subset of AMD clinic patients was merged with the ZVF 12 month visual function data subset. Thus SD OCT scans of these 29 Clinic subjects, 50 to 88 years (73 ± 10.5 years) of age with mild atrophic AMD cases (but without vision function data) were merged with the ZVF subjects to create a second larger data set consisting of 98 eyes of fifty four subjects (49 males and 5 females) with an age range of 50 to 91 years (74 ± 10.1 years). Only patients with atrophic AMD and typical SD OCT retinal signs were included. For example, RPE/Photoreceptor or photoreceptor integrity line (PIL) disruption, soft drusen, drusenoid RPE detachments, outer nuclear layer reflective (inflammatory) bodies, *etc.* but without evidence of sensory retinal detachment or sub-retinal neovascularization [6]. Subjects with the entity "Age-Related Choroidal Atrophy" manifesting extreme choroidal thinning of less than 125 microns were excluded [11]. Potential retinal thickness and visual function confounders, such as subjects with glaucoma, significant diabetic retinopathy (*i.e*., macular edema), pre-retinal gliosis and vitreal traction syndrome were excluded. In the ZVF group, four patients had both eyes and three patients had only one eye excluded whereas in the eye clinic AMD patient group, seven

patients had both eyes excluded. Subjects were limited to ±6.00 DS refractive error. Informed

consent was obtained from ZVF subjects and the Declaration of Helsinki protocols were maintained. The RTVue provides a detailed sampling of over 19,000 thickness points plotted in a 5 mm × 5 mm area of the central macula. The Full Thickness MM5 Significance Map presentation reveals the significance of the full retinal thickness deviation from normal, of the scanned retina (Figure 1a). The total area of significant loss (blue areas in the Significance Map) was calculated and divided by the total area of the Map to calculate the percent extrafoveal thinning (% EFT), and used for functional correlation. The MM5 provides nine retinal thickness measurements based on the ETDRS, is centered on the fovea with a diameter of 1 mm, four *parafoveal quadrants* surrounding the central fovea extending form 1 mm out to 3 mm, and four perifoveal *quadrants* extending from 3 mm to 5 mm. Extra foveal paramaters refer to the sum of *parafoveal* + *perifoveal* area. Figure 2 depicts these nine regions**.** An age-matched subset of the RTVue normative database was used for comparison. Normative data from 309 patients (594 retinas) age 50–82 were available. The subfoveal choroidal thickness was measured using the En Face viewing option of the 3D macular presentation. This view presents the sum of all c-scan planes of the same 5 mm macular radius in a top-down fundus view. The choroidal layer was determined to be the distance between Bruch's membrane to a depth at which choroidal vessels can no longer be seen. Two measurements were made for each eye, with an acceptable intra-observer (author JC) coefficient of variation of +0.90. The average choroidal thickness of these two measurements is reported.

Visual psychophysical data, lens opacification and NEI VFQ25 vision questionnaire and retinal grading were available for ZVF patients only. Best refracted distance visual acuity measured with randomly presented ETDRS letters on an M&S Technologies (Chicago, IL) SmartSystem II LCD monitor viewed at 10 feet (SR). The CSF test was performed using best refracted maximum visual acuity and the Functional Vision Anayzer (Stereo Optical, Inc Chicago, IL) as previously described [12,13]. Blue/Yellow Color Vision increment thresholds were ascertained with the ChromaTest® (Bromley, UK) with best corrective lenses according to protocol [10,14,15]. Integrated 6 degree diameter Macular Pigment (MP) area under the curve (AUC) was determined objectively with an ARIS 110 camera (Visual Pathways, Prescott, AZ) by the method of specular reflectance [16].

**Figure 2.** Full thickness SD OCT MM5 EDTRS significance map schematic. Nine zone ETDRS (Early Treatment of Diabetic Retinopathy) grid provided by the RTVue SDOCT measures an area 5 mm × 5 mm on the retina (MM5 retinal thickness scan pattern). This grid is centered on the fovea and shows the relative location of each of four extra-foveal quadrants defined as the sum of the parafovea + perifovea regions (figure courtesy Optovue Inc., Fremont, CA).

Lens opacification, a confounder of CSF, was qualitatively assessed using a seven increment scaled LOCSIII transparency with multivariate analysis as previously described [9,12,13]. Fifty degree fundus images were taken with a Kowa VX-10 (Tokyo, Japan). Retinal grading of AMD disease state was completed in a double-masked randomized fashion by a retinal specialist (ML) utilizing the AREDS report #18 simplified (0–4) scale for presence of hypo/hyper RPE pigmentation and soft drusen [17]. Overall visual function was assessed with the NEI VFQ25 vision function questionnaire [10].

Statistics: As with ZVF [10], average-eye visual function data from a single subject, and significance, is typically shown for simplicity, in cases in which the analysis did not differ appreciably from evaluating right eyes and left eyes separately. Thus the last three figures in the RESULTS section present data for both eyes. Statistical analysis utilized two sided, student *T*-tests for paired comparison data and *P* < 0.05 was considered significant.

#### **3. Results**

The right, left and average eye AMD simplified AREDS grade retinopathy for the ZVF fundus image data set was 0.71 (SD 0.6)/2.0 max score, 0.71 (SD 0.5) 2.0 max score, and 1.43 (SD 0.92)/ 4.0 max bilateral retinal score signifying mild AMD. Average eye ETDRS visual acuity was 20/20-1 (SD 1.2 lines). Figure 3 is a scatter plot of age *vs.* average foveal thickness from *n* = 594 "Normal" retinas from patients ages 50–82 of various ethnicities. This Optovue normative data suggests a slight foveal thickening with age (*r* = 0.09, *P* = 0.02). AMD patients from the ZVF study (*n* = 29 AMD retinas) and the larger merged clinical/research AMD data set (*n* = 54 AMD retinas), ages 50+ reveal: (1) slightly thinner foveal thickness compared to these "Normals" and (2) a thickness that does not change with age. Regardless, "Normal" and AMD foveal thickness appears to be highly guarded, conserved, and little changed with age.

**Figure 3.** Foveal thickness and aging. RTVue SD OCT normative retinal thickness MM5 data from (*n* = 597) international retinas of an all ethnicities and gender data base between the ages of 50 and 82. The normal fovea thickens approximately 2.5 ȝm/decade. (Courtesy Optovue, Fremont, CA) Also shown is retinal thickness *vs.* age for post study ZVF AMD subjects (*n* = 29 retinas) and a combined AMD clinical patient population inclusive of these research (*n* = 54 retinas) with a greater number of AMD subjects in the 82–91 age group.

In contradistinction to this slight foveal thickening, Figure 4a–d and Table 1 show parafoveal + perifoveal quadrants slightly *thin* with age in normals, although not significantly: (Superior Quadrant: í1.1 ȝm/decade; Nasal Quadrant: í1.0 ȝm/decade; Inferior Quadrant í0.6 ȝm/decade and Temporal Quadrant í0.3 ȝm/decade). In both the ZVF AMD study and the larger merged AMD clinical data set, overall macular thickness excluding the fovea show accelerated thinning at a rate of í6.7 ȝm/decade (*P* = 0.03) and í4.6 ȝm/decade (*P* = 0.04) respectively. Specifically, AMD patients thin by í5.5 ȝm/decade (*P* < 0.02) superiorly, í5.0 ȝm (*P* < 0.04) nasally, í4.5 ȝm (*P* < 0.07 for trend) inferiorly, and í3.3 ȝm (NS) temporally using data from 54 AMD retinas. Table 1 summarizes this AMD retinal thinning effect. This difference is outside the 5 ȝm coefficient of variation of the instrument. More specifically, Table 2 depicts accelerated thinning of macular retinal thickness (ȝm/decade) in the overall 5 mm retinal diameter ETDRS diameter grid *excluding the fovea* and extrafoveal quadrants (but not the fovea) in AMD retinas, compared with the age matched "Optovue normative" population data. A statistically significant AMD retinal thinning effect is particularly evident in the vulnerable superior retinal ETDRS quadrant compared with the temporal quadrant (ns).

**Figure 4.** Extra-foveal quadrant thickness *vs.* age. (**a**) Superior quadrant; (**b**) Nasal quadrant; (**c**) Inferior quadrant; (**d**) Temporal quadrant. Plots reflect the RTVue SD OCT age matched normative data against the two AMD retina data sets: ZVF and the larger clinical AMD patient data set that includes ZVF study patients. Both AMD data sets display accelerated thinning with age, especially for the superior quadrant (*P* < 0.02). See Table 1.

**Figure 4.** *Cont.*

(**d**)

**Table 1.** The mean (SD) retinal thickness (m) of patients with AMD is thinner than those from the OptoVue SD OCT "normative" population data base. This difference is outside the 5 m axial published instrument coefficient of variation.

**Table 2.** Accelerated thinning of macular retinal thickness (ȝm/decade) in the overall 5 mm retinal diameter ETDRS diameter grid *excluding the fovea and* extrafoveal quadrants (but not the fovea) in ZVF Study and clinical AMD retinas, compared with the age matched "Optovue normative" population data. A statistically significant thinning effect in AMD is particularly evident in the vulnerable superior retinal parafoveal + perifoveal ETDRS quadrant [10], see text.

\* Indicates significant data; 1 Optovue, Inc., Fremont, CA, normative database.

Figure 5 depicts loss of CSF with age for the ZVF AMD subjects. Parenthetically, the NEI Vision Function Questionnaire (VFQ 25) total all category summed score (data not shown) was positively correlated with this loss of contrast sensitivity (*r* = +0.27, *P* = 0.04) and inversely correlated with % total retinal thinning but not significantly (*r* = í0.20, *P* = 0.13). As shown in Figure 6a, CSF is inversely related to % macula thinning (*r* = í0.33, *P* = 0.01, 2 Tailed T Paired Comparison) , while Figure 6b depicts the 6.5 degree ChromaTest® B/Y threshold to be heightened and directly related to % macula thinning (*r* = +0.34, *P* = 0.01, 2 Tailed T Paired Comparison).

Retinal thickness (excluding the fovea) correlated with choroidal thickness, but not significantly (*R* = +0.15, *P* = 0.13). However, Figure 7a shows a significant negative correlation between % thinning and choroidal thickness (*r* = í0.25, *P* = 0.01) for all subjects while Figure 7b suggests that parafoveal/perifoveal thinning is associated with declining overall volumetric macular pigment optical density (MPOD), but not significantly (*r* = í0.21, *P* = 0.14). Precise three dimensional MPOD data was available only for ZVF subjects.

**Figure 5.** Contrast sensitivity *vs.* age for ZVF AMD subjects. Loss of CSF with age in *n* = 56 AMD eyes from the ZVF study. Integrated area under the curve of the contrast sensitivity function (CSF) at four spatial frequencies: 3 cc, 6 cc, 12 cc and 18 cc/degree [10]. (Functional Vision Analyzer®, Stereo Optical, Inc., Chicago, IL).

**Figure 6.** The visual consequences of extra-foveal thinning. (**a**) Contrast sensitivity (area under the curve) is inversely related to % retinal thinning (*r* = í0.33 *P* = 0.01, 2 Tailed paired *T* test). (**b**) 6.5 degree increment blue-threshold is directly related to % retinal thinning (*r* = +0.34, *P* = 0.01, 2 Tailed paired *T* test). Blue/Yellow Color Vision increment thresholds were ascertained with the ChromaTest® (Bromley, UK) with best corrective lenses according to protocol [10,14,15].

**Figure 6.** *Cont.*

**Figure 7.** Retinal and choroidal thinning are associated. (**a**) SD OCT measure of choroidal thickness *vs.* % retinal thinning. The coefficient of variation with the manual caliper method, same observer, was *r* = 0.90. Note the significant negative correlation between % retinal thinning and choroidal thickness (*r* = í0.25, *P* = 0.01) for all subjects. (**b**) Three dimensional macula pigment optical density (integrated volume) specular reflectance measurement (ARIS 110 camera, Visual Pathways, Prescott, AZ) *vs.* retinal thinning, reveals % extrafoveal thinning to be associated with declining volumetric macular pigment optic density (MPOD), but not significantly.

#### **4. Discussion**

Foveal thickness appears well protected with age. Even our early AMD patients had no change in central foveal thickness with age. One might expect one or more of the following factors to be associated with age-related retinal thinning: (1) retinal atrophy; (2) choroidal atrophy; or (3) lower foveal macular pigment. We believe the third explanation is germane to this report, as all of the subjects, both post ZVF Study subjects and clinical patients, despite their advancing age had supplemented with approximately 10 mg or more of the oral retinal carotenoids lutein and/or zeaxanthin.

In a manner analogous to neural retinal rim thinning or more recently the concept of pre-glaucomatous ganglionic layer (GCC) cell loss in glaucoma, the atrophic extrafoveal macula appears to thin in AMD at a greater rate than predicted by simple aging. Confounding high myopia was excluded from our database. The order effect with superior most vulnerable and temporal least vulnerable was shown in two data sets—a research data set and a merged research + clinical data set. This pathological patterned loss of tissue was first eloquently revealed by Sarks, Sarks and Killingsworth decades ago using serial fluorescein angiography, white light, and electron microscopy [18]. Our results agree with that historic study.

Our observation that atrophic AMD typically begins in the superior parafovea (at approximately 7 degrees eccentricity) has been confirmed by two other research groups and at the ARVO 2010 meeting (Ft Lauderdale, FL). Significantly, at the Oregon National Primate Research Center, Rhesus monkeys deprived of carotenoids from birth were found to develop atrophic AMD in an extra-foveal superior location [19]. A second group from the Australian National University, using a novel pupillary reaction test, mapped human AMD functional deficits in sector pupillary latency to the superior parafovea [20]. Furthermore, it has been suggested, that the superior half of the retina is vulnerable due to the damaging effects of reflective UV ambient ground reflecting radiation experienced by humans as primates typically walk with a slight head-down gaze (Personal Communication [21]). Ascertaining the health of the superior parafoveal rod rich retina is the basis of the new Maculogix (Atlanta, GA) AMD detection device.

A limitation of this study is the dearth of "normative" patients in the age 82–91 range (Figure 3) combined with the high likelihood of unrecognized subclinical AMD in the "normative" data base especially in old old "normals". However, this lack of patients in the oldest age group combined with the fact that our AMD subjects all had visible retinal pathology creates a counterbalancing bias. This report further supports the conclusion of both the Lutein Antioxidant Supplementation Trial (LAST) and recently published ZVF Study, that subtle signs of photoreceptor—retinal pigment epithelium disturbance characteristic of AMD, such as CSF and increment B/Y color vision thresholds, are impaired long before the appearance of obvious ophthalmoscopic or nonspectral photographic AMD pathological signs appear (*i.e*., soft drusen or RPE rarefaction). That is, functional visual decline *precedes* photoreceptor—retinal pigment epithelium atrophy [10,12]. The AMD patients in this report had early disease with an AREDS Report #18 five year risk estimate of only 8% for future catastrophic monocular vision loss, near perfect 20/20 ETDRS visual acuity, and no fovea retinal thinning.

In our combined clinical and research experience, patients with even extreme extra-foveal retinal thinning might go undetected in current clinical practice where CSF or even simple, quick near point low contrast Colenbrander visual acuity screening is seldom utilized except perhaps in refractive surgical and contact lens practices [22]. It makes sense that large field (6.5 degree) blue sensitivity suffers earliest in AMD because blue cones are fragile and few in number [14]. By the time the patient is in danger of an acute degenerative change, a flashed 6.5 degree diameter blue/yellow stimulus optotype becomes indistinguishable no matter how intense the color. Once again, subtle color vision disturbances in early AMD are rarely evaluated except in the research laboratory. The decline in visual function in early AMD is far from academic. CSF is related to driving safety and hip fracture risk as well as overall quality of life with mortal implications [23–25]. While a 10-letter (two-line) decrease in best corrected VA is significantly associated with all self-reported measures of visual disability, a mere 2-step decrease in contrast sensitivity is significant [25].

There is increasing recognition that the optical and antioxidant properties of the xanthophyll pigments lutein, zeaxanthin, and lutein's metabolite mesozeaxanthin, play an important role in maintaining the health and function of the human macula [16]. Foveal MPOD has a significant positive, measurement technique independent relationship of approximately *r* = +0.30 with central retinal thickness [26]. Significantly, in that study there was no demonstrable relationship between MPOD at an eccentricity of one or two degrees and central retinal thickness. However in the present AMD study, Figure 7b shows a non-significant trend for extra-foveal thinning to be associated with declining volumetric macular pigment. We also found no significant change in central foveal thickness in agreement with the excellent preservation of visual acuity in both our research and clinical population. Interestingly, several groups have established that MPOD tends to decrease with age [16]. This is conjectured to contribute to the accelerated retinal thinning we observed ostensibly through either direct deposition (*i.e*., de-pigmentation) or by secondary indirect declining antioxidant tissue protection of the retina and the choriocapillaris/choroidal vascular bed [27]. We also found a significant association between choroidal thinning and retinal thinning. Spaide describes a decrease of 16 ȝm/decade choroidal thickness in high myopes [11]. Our ZVF AMD patients showed an even greater 22 ȝm/decade decrease in choroidal thickness with each decade (*r* = í0.57; *P* < 0.01).

#### **5. Conclusion**

New technologies and techniques are required to evaluate the effect of nutrient intervention on retinal health in early and moderate AMD. In this report, we present a new objective SD OCT based atrophic AMD metric called "% extra-foveal retinal thinning" or % EFRT. We have shown that % EFRT is associated with loss of contrast sensitivity and extra-foveal blue cones in early AMD. Macular thickness ETDRS SD OCT thickness plots provide eye practitioners with a new clinically useful and objective surrogate measure of early functional visual loss of aging retinas. Indeed, the data and simulations presented in this report explain why practitioners often underestimate the functional impact of a "20/20 AMD diagnosis". Researchers can employ CSF/Low Contrast Screening, SD OCT and MP distribution instrumentation together, to gauge overall and distributional changes in retinal thickness and formally evaluate how prescriptive carotenoid repletion (*i.e*., lutein/mesozeaxanthin and zeaxanthin along with synergistic *n*-3 fats) impacts % EFRT and visual function. This new metric can be applied to the study of aging, genetic susceptibility and age related choroidal degeneration. The authors believe % EFRT to be superior to the following three currently employed tests: the 150 year old Snellen chart, the 117 year old Amsler Grid and conventional non-spectral fundus cameras. Significantly, % EFRT has potential to meaningfully and efficiently evaluate both nutritional and pharmacologic atrophic AMD intervention(s) in clinical practice as well as retinal research.

#### **Acknowledgements**

This material is based on original work supported by the James A Lovell Federal Health Care Center, Eye Clinic (North Chicago, IL) and the Department of Veterans Affairs Research Service/CARES (Hines, IL). Chrysantis/Ball Horticulture (West Chicago, IL) and Kemin Health LC (Des Moines, IA) were the primary granting sponsors of ZVF and LAST respectively. Stereo Optical, Inc. (Chicago, IL) and Optovue (Fremont, CA) provided instrumentation as secondary sponsors of ZVF: FDA IND #78, 973 and our Ocular—Nutrition Laboratory.

#### **Conflict of Interest**

The authors declare no conflict of interest. Dr. Richer is a consultant to Stereo Optical, Inc.

#### **References**

1. Smith, W.; Assink, J.; Klein, R.; Mitchell, P.; Klaver, C.C.; Klein, B.E.; Hofman, A.; Jensen, S.; Wang, J.J.; de Jong, P.T. Risk factors for age-related macular degeneration: Pooled findings from three continents. *Ophth*almology **2001**, *108*, 697–704.

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**6. The use of Prolonged CS Exposure (PE)**

The technique of prolonged CS exposure was first developed by Thomas G. Stampfl in the late 1950's. Despite considerable pressure to publish, he refused to publish until two experimental outcome studies were conducted to support his clinical finding. They were provided by Hogan (1966) who used a psychiatric hospitalized population, and Levis & Carrera (1967) who used an outpatient population. He then published his first article (Stampfl & Levis, 1967). He labeled his new response prevention approach, Implosive Therapy (IT). He first used an in vivo extinction approach but changed to an imagery approach due to the ability to introduce hypothesized conditioned stimulus (CS) that do not lend themselves readily to in vivo exposure (e.g., the fear of bodily injury). Stampfl adapted

A Review of Childhood Abuse Questionnaires and Suggested Treatment Approaches 15

Bernstein, D.P., Fink, L., Handelsman, L., Foote, J., Lovejoy, M., Wenzel, K., Sapareto, E., &

child abuse and neglect. *American Journal of Psychiatry*, 151 (8), 1132-1335. Bernstein, D.P., Abuvalia, T., Pozzi, D., & Handelsman, L. (1997). Validity of the Childhood

Boudewyns, P.A., & Levis, D.J. 91975). Autonomic reactivity of high and low ego-strength

Boudewyns, P.D., & Shipley, R.H. (1983). *Flooding and Implosive Therapy*. New York: Plenum

Boyd, T.L., & Levis, D.J. (1980). Depression: In Dartzman, R.J. (Eds.), *Clinical Behavior Therapy and Behavior Modification*, New York: Garland STPM Press, pp. 301-349. Brandyberry, L.J., & McNair-Semands, R.R. (1998). Examining the validity and reliability of

Briere, J. (1989). Therapy for adults molested as children: Beyond survival. New York, New

Briere, J. (1992b). Methodological issues in the study of sexual abuse effects. *Journal of*

Briere, J. (1995). Trauma Symptom Inventory: Psychometrics and association with childhood

Briere, J., Evans, D., Runtz, M., & Wall, T. (1988). Symptomatology in men who were

Briere, J., & Runtz, M. (1989). The Trauma Symptom Checklist (TSC-33). *Journal of*

Briere, J., & Conte, J. (1991). Self-reported amnesia for abuse in adults molested as children.

Briere, J., Elliott, D.M., Harris, K., & Cotman, A. (1994). The trauma symptom inventory:

Briere, J., & Runtz, M. (1986). Suicidal thoughts and behaviours in former sexual abuse

Briere, J., & Runtz, M. (1990). Differential adult symptomatology associated with three types

Briere, J. & Elliott, D.M. (1997). Psychological assessment of interpersonal victimization

Brown, J. (1990). The treatment of male victims with mixed=-gender, short-term group

Browne, A., & Finkelhor, D. (1986).  In M. Hunter (Ed.) *The Sexually Abused Male*, Vol. 2. Lexington:

Southern California School of Medicine, Los Angeles, CA.

victims. *Canadian Journal of Behavioural Sciences*, 18, 413-423.

of child abuse histories. *Child Abuse and Neglect*, 14, 357-364.

effect in adults and children. *Psychotherapy*, 34 (4) 353-364.

*American Academy of Child in Adolescents Psychiatry*, 36 (3), 340-348.

692.

Press.

387-401.

(3), 457-461.

Lexington Books.

*Neglect,* 22 ( 12), 1253-1263.

York: Springer Publishing Company.

*Interpersonal Violence*, 4, 151-163.

*Journal of Traumatic Stress*, (6), 21-31.

*Psychological Bulletin*, 99 (1), 66-77.

Briere, J. (1992a). *Child Abuse Trauma*. Newbury Park: Sage Publications.

*Consulting and Clinical Psychology*, 60 (2), 196-203.

Ruggerio, J. (1994). Initial reliability and validity of a new retrospective measure of

Trauma Questionnaire in an adolescent psychiatric population. *Journal of the*

subjects to repeated anxiety eliciting scenes. *Journal of Abnormal Psychology*, 84, 682-

childhood abuse scales: Putting the courage to heal to the test. *Child Abuse &*

and adult victimization in clinical samples. *Journal of Interpersonal Violence*, 10 (4),

molested as children: A comparison study. *American Journal of Orthopsychiatry*, 58

Reliability and validity in a clinical sample. Unpublished manuscript, University of

a revision of Mowrer's two factor avoidance theory (see Levis, 1979). His theory (Stampfl, 1970, Levis, 1985) has received strong experimental support at the human and infrahuman level of analyses (Levis, 1979, 1985) including a resolution of Freud's neurotic paradox as to why clinical symptoms persist over time (Levis & Brewer, 2001). The technique has been successfully used in the treatment of obsessive compulsive behavior (Foa, 2000); in the treatment of panic disorders (Levis, 1987); phobic behavior (Stampfl & Levis, 1967) and depression (Boyd & Levis 1980). The IT technique is an operational design feedback technique capable of re-activating trauma memories. The frequent use of Stampfl's therapist directed imagery technique lead to the development of a free-recall technique (referred to as patient directed IT or brain release therapy). This approach appears capable of providing a complete re-activation of trauma experiences (Levis, 1988; Levis 1995; Kirsch & Levis, 2001).

Today, prolonged exposure therapy is perhaps the most frequent and empirically supported technique for use for patient diagnosed with PTSD including survivors of childhood abuse (see Morrison, 2011). Yet as Morrison regrettably notes, some therapists avoid its use. Part of the concern is related to the fear that the high levels of anxiety elicited may be harmful to the patent. This fear exists despite the strong experimental evidence that the approach is not harmful (see Boudewyns & Levis, 1975; Boudweyn & Shipley 1983). Another key factor in the avoidance of using PE relates to the emotional impact on the therapist who witnesses the patient's trauma as they re-experience their historical emotional trauma. Despite this difficulty, I have found the law of extinction will eventually work for both the patient and the therapist resulting in a strong positive reinforcement when a reduction for symptomatology occurs. It has also been my experience that trauma victims rarely terminate therapy prematurely or miss a session.

In closing, it is my hope this chapter will not only alert the reader to the frequent and devastating effects that childhood abuse has on the adult survivors of abuse but also serve as an alert to practitioners as to the importance of providing a comprehensive assessment of the potential presences of an abusive childhood history in their patients.

#### **7. References**

a revision of Mowrer's two factor avoidance theory (see Levis, 1979). His theory (Stampfl, 1970, Levis, 1985) has received strong experimental support at the human and infrahuman level of analyses (Levis, 1979, 1985) including a resolution of Freud's neurotic paradox as to why clinical symptoms persist over time (Levis & Brewer, 2001). The technique has been successfully used in the treatment of obsessive compulsive behavior (Foa, 2000); in the treatment of panic disorders (Levis, 1987); phobic behavior (Stampfl & Levis, 1967) and depression (Boyd & Levis 1980). The IT technique is an operational design feedback technique capable of re-activating trauma memories. The frequent use of Stampfl's therapist directed imagery technique lead to the development of a free-recall technique (referred to as patient directed IT or brain release therapy). This approach appears capable of providing a complete re-activation of trauma experiences (Levis, 1988; Levis 1995; Kirsch & Levis, 2001). Today, prolonged exposure therapy is perhaps the most frequent and empirically supported technique for use for patient diagnosed with PTSD including survivors of childhood abuse (see Morrison, 2011). Yet as Morrison regrettably notes, some therapists avoid its use. Part of the concern is related to the fear that the high levels of anxiety elicited may be harmful to the patent. This fear exists despite the strong experimental evidence that the approach is not harmful (see Boudewyns & Levis, 1975; Boudweyn & Shipley 1983). Another key factor in the avoidance of using PE relates to the emotional impact on the therapist who witnesses the patient's trauma as they re-experience their historical emotional trauma. Despite this difficulty, I have found the law of extinction will eventually work for both the patient and the therapist resulting in a strong positive reinforcement when a reduction for symptomatology occurs. It has also been my experience that trauma victims rarely terminate

In closing, it is my hope this chapter will not only alert the reader to the frequent and devastating effects that childhood abuse has on the adult survivors of abuse but also serve as an alert to practitioners as to the importance of providing a comprehensive assessment of

Agor, K., Read, J., & Bish, J. (2002). Identification of abuse histories in a community mental

Alexander, F. (1965). The dynamics of psychotherapy in light of learning theory.

Allen, J.G., Huntoon, J., & Evans, R.B. (1999). Self-report measures to screen for trauma

Astin, M.C., Lawrence, K.J., & Foy, D.W. (1993). Post-traumatic stress disorder among battered women: risk and resiliency factors. *Violence and Victims*, 8, 17-29. Beck, A.T., Steer, R.A., & Garbin, M.G. (1988). Psychometric properties of the Beck

Bell, J., & Belichi, K. ( 1998). A community study of well-being in adults reporting childhood

health center: The need for policies and training. *Journal of Mental Health (UK)*, (11),

history and its application to women in in-patient treatment for trauma related

Depression Inventory: Twenty-five years of evaluation. *Clinical Psychology Review*,

the potential presences of an abusive childhood history in their patients.

*International Journal of Psychiatry*, I, 189-97.

disorders. *Bulletin of the Menninger Clinic*, 63(3).

abuse. *Child Abuse and Neglect*, 22(7), 681-685.

therapy prematurely or miss a session.

**7. References**

533-543.

8, 77-100.

A Review of Childhood Abuse Questionnaires and Suggested Treatment Approaches 17

Feldman-Summers, S., & Pope, K.S. (1994). The experience of "forgetting" childhood abuse:

Fergussen, D.M., Lynskey, M.T., & Harwood, L.J. (1996). Childhood sexual abuse and

Finkelhor, D., Hotaling, G., Lewis, I.A., & Smith, C. (1990). Sexual abuse in a national survey

Finestone, H.M., Stenn, P., Davies, F., Stalker, C., Fry, R., Koumanis, J. (2000). Chronic pain

Foa, E.B. (2000). Psychosocial treatment of post-traumatic stress disorder. *Journal of Clinical*

Gelles, R.J., & Straus, M.A. (1987). Is violence toward children increasing? A comparison of 1975 and 1985 national survey rats. *Journal of Interpersonal Violence,* 2 (2), 212-222. Gold, S.R., Milan, L.D., Mayall, A., & Johnson, A.E. (1994). A cross-validation study of the

Gold, J.W., & Cardena, E. (1998). Convergent validity of three post-traumatic symptom

Goldwater, L., & Duffy, J.F. (1990). Use of the MMPI to uncover histories of childhood abuse in adult female psychiatric patients. *Journal of Clinical Psychology*, 46, 392-398. Hathaway, S.R., & McKinley, J.C. (1989). *The Minnesota Multiphasic Personality Inventory II*,

Herman, J.L., & Schatzow, E. (1987). Recovery and verification of memories of childhood

Hogan, R.A. (1966). Implosive therapy in the short term treatment of psychotics.

Jacobson, A. (1989). Physical and sexual assault histories among psychiatric out-patients.

Jacobson, A. (1990). The relevance of childhood sexual abuse to adult psychiatric in-patient

Kean, T.M., Malloy, P.F., & Fairbank, J.A. (1984). Empirical development MMPI subscale for

Kendall-Tackett, K.A. (2000). Physiological correlates of childhood abuse: chronic

Kent, A., & Waller, G. (1998). The impact of childhood emotional abuse: An extension of the child abuse and trauma scale. *Child Abuse and Neglect*, 22 (5), 393-399. Kilpatrick, D.A. (1988). Rape aftermath symptom test. In M. Herson & A.S. Bellack (eds.) *Dictionary of Behavioral Assessment Techniques*, New York: Pergamon Press.

the assessment of combat-related post-0traumatic stress disorder. *Journal of*

hyperarousal in PTSD, depression and irritable bowel syndrome. *Child Abuse &*

636-639.

*Psychiatry*, 34 (10).

*and Neglect*, 14, 19-28.

*Psychiatry*, 61, 43-53.

*Violence, 9, 12-26.*

*Child Abuse & Neglect*, 24 (5) 547-556.

 *Psychoanalytic Psychology*, 4 (1), 1-14.

*Psychotherapy, Theory, Research & Practice*, 25-31.

care. *Hospital Community Psychiatry*, 41 (7), 807.

*Consulting and Clinical Psychology*, 52, 888-891.

*Neglect*, 24 (6), 799-810.

*American Journal of Psychiatry*, 146, 755-758.

A national survey of psychologists.  *Journal of the American Academy of Child and Adolescent*

of adult men and women: Prevalence, characteristics and risk factors. *Child Abuse*

and health care utilization in women with a history of childhood sexual abuse.

trauma symptom checklist: The role of mediating variables. *Journal of Interpersonal*

inventories among adult abuse survivors. *Journal of Traumatic Stress*, 11 (1), 173-180.

Buchanan, A., & Oliver, J.E. (1977). Abuse and neglect as a cause for retardation. *British*

Bursten, B. (1995). Validity of child abuse measurements [letter to the editor]. *American*

Castelda, B.A. (2003). An examination of the Sexual Questionnaire-III in identifying self-

Castelda, B.A. (2006). Construct validation of the Sexual Abuse Questionnaire (SAQ) using

Castelda, B.A., Levis, D.J., Rourke, P.A.,. & Coleman, S.L. (2007). Extension of the Sexual

Coverdale, J.H., & Turbott, S.H. (2000). Sexual and physical abuse of chronically ill

Curtis, L.A. (1976). Present and future measures of victimization in forcible rape. In M.J.

Daro, D. (1988). Confronting child abuse: Research for effective program design. New York,

Demause, L. (1991). The Universability of incest. *The Journal of Psychohistory*, 19 (2), 123-164. DeVellis, R.F. (1991). *Scale Development: Theory and Application*. Newbury Park: Sage

Dinwiddie, S., Heath, A.C., Dunne, M.P., Bucholz, K.K., Madden, P.A.F., Slutski, W.S.,

psychopathology: A co-twin-control study. *Psychological Medicine*, 30, 41-52. Domino, J.V., & Haber, J.D. (1987). Prior physical and sexual abuse in women with chronic

Doyle-Peters, S., Wyatt, G.E., & Finkelhor, D. (1986). Prevalence. In D. Finkelhor (ed.) *A Sourcebook on Child Sexual Abuse*. Beverly Hills, CA: Sage Publications. Edens, J.F., Otto, R.K., & Dwyer, T.J. (1998). Susceptibility of the Traumatic Symptom Inventory to malingering. *Journal of Personality Assessment*, 71 (3), 379-392. Elliot, D.M., & Briere, J. (1990, April). *Predicting molestation history in professional women with*

Elliot, D.M., & Briere, J. (1991). Studying the long-term effects of sexual abuse: The trauma

Elliott, D.M., & Briere, J. (1992). Sexual abuse trauma among professional women:

reported childhood sexual, physical, and combined abuse, unpublished master

an acoustic startle procedure and measures of complex Post-Traumatic Stress Disorder (PTSD), unpublished doctoral dissertation, Binghamton University,

Abuse Questionnaire to other abuse categories: The initial psychometric validation of the Binghamton Abuse Screen*, Journal of Child Sexual Abuse*, (16), 1, 207-325. Coleman, S.L., Rourke, P.A., & Levis, D.J. (2008). Experimental and self-reported indices of

childhood abuse using the Binghamton Childhood Abuse Screen. *Journal of*

psychiatric outpatients compared with a matched sample of medical outpatients.

Walker & S.L. Brodsky (eds.) *Sexual Assault: The victim and the rapist*. Lexington,

Bierut, J.J., Statham, D.B., & Martin, N.G. (2000).  Paper presented at the annual meeting of

symptom checklist (TSC) scales. In A.W. Burgess (ed.), *Rape & Sexual Assault III: A*

Validating the trauma symptom checklist-40 (TSC-40). *Child Abuse & Neglect*, 16 (3),

*Journal of Psychiatry*. 131, 459-467.

Binghamton, New York, USA.

MA: Lexington Books.

Publications.

391-398.

New York. The Free Press.

*Journal of Psychiatry*, 152 (10), 1533-1534.

thesis, Binghamton University, Binghamton, NY, USA.

*Aggression, Maltreatment & Trauma*, 17 (4), 506-619.

*Journal of Nervous and Mental Disorders*, 188 (7), 440-5.

headache: Clinical correlates. *Headache*, 27, 130-134.

the Western Psychological Association, Low Angeles.

*Research Handbook* (pp. 55-74), New York, NY: Garland.

A Review of Childhood Abuse Questionnaires and Suggested Treatment Approaches 19

Lipschitz, D.S., Bernstein, D.P., Winegar, R.K., & Southwick, S.M. (1999). Hospitalized

Lock, T.G. (1997). Development of the Sexual Abuse Questionnaire: An examination of

Lock, T.G., Levis, D.J., & Rourke, P.A. (2005). The Sexual Abuse Questionnaire: A

Lothian, J., & Read, J. (2002). Asking about abuse during mental health assessments: Clients' views and experiences. *New Zealand Journal of Psychology*, 31, 98-103. Lynch, D., Lowe, R.R., & McKenna, J. (2010). Cognitive behavioral therapy for major

Macmillian, M., Flemming, J., Trocme, N., Boyle, M.H., Wong, M., Racine, O.A., Beardslee,

Malinosky-Rummell, R., & Hansen, D.J. (1993). Long-term con sequences of childhood

Martin, J., Anderson, J., Romans, S., Mullen, P., & O'Shea, M. (1993). Asking about child

Morrison, J.A. ( 2011 ) The Therapeutic relationship in prolong exposure therapy for post-

Mullen, P,.E., Martin, J.L., Anderson, J.C., Romans, S.E., & Herbison, G.P. (1996). The long-

Perkins, D.F., Luster, T., & Jank, W. (2002). Protective factors, physical abuse, and purging

Polusny, M.A., & Follette, V.M. (1995).  Assessment of rape-related post-traumatic

Robbins, A.D. (1995). False memories or hidden agendas? *Journal of Psychohistory*, 27 (3),

Rodriguez, N., Ryan, S.W., Rowan, A.B., & Foy, D.W. (1996). Post-traumatic stress disorder

reported measures. *Journal of Traumatic Stress*, 12 (4), 641-654.

controlled trials. *Psychological Medicine*, 40, 9-24.

physical abuse. *Psychological Bulletin*, 114 (1), 68-79.

community study. *Child Abuse & Neglect*, 20 (1), 7-21.

*Medical Association*, 278 (2), 131-135.

*Behavior Therapist*. Vol.34, No.2.

*Neglect*, 17, 383-392.

17 (4), 377-400.

166.

572.

305-311.

*Neglect*, 20 (10), 943-952.

 Unpublished master thesis, Binghamton

preliminary examination of a time and cost efficient method of evaluating the presence of childhood sexual abuse in adult patients. *Journal of Child Sexual Abuse*,

psychiatric disorders: Does it really work? A meta-analytical review of well

W.R., & Offord, D.R. (1997). Prevalence of child physical and sexual abuse in the community, results from the Ontario Health supplement. *Journal of the American*

sexual abuse: Methodological implications of a two stage survey. *Child Abuse &*

traumatic stress disorder: The role of cross-theoretical dialogue in dissemination.

term impact of the physical, emotional, and sexual abuse of children: A

from community-wide surveys of female adolescents. *Journal of Adolescent Research*,

and review of the empirical literature. *Applied and Preventative Psychology*, 4, 143-

stress disorder: Stressor and symptom dimension.  *Child Abuse &*

Kirsch, M.V., & Levis, D.J. (2001). An assessment of the accuracy of two clinical procedures

Krantweiss, A. (2001). The utility of video film clips of sexual abuse in assessing SAQ-III and

Krantweiss, A. (2004). The utility of video film clips of sexual and physical abuse in

Layman, M., Gidycz, C., & Lynn, S.J. (1996). Unacknowledged versus acknowledged rape

Lesserman, J., Drossman, D.A., Li, Z., Toomey, T.C., Nachman, G., & Glogau, L. (1996).

Lenzenweger, M.F., Loranger, A.W., Korfine, L., & Neff, C. (1997). Detecting personality disorder in a non-clinical population. *Archives of General Psychiatry*, 54 (4), 345-351. Levis, D.J. & Boyd, T.L. (1979). Symptom maintenance: An infrahuman analysis and

Levis, D.J. (1985). Implosive theory: A comprehensive extension of conditioning theory of

Levis, D.J. (1987). Treating anxiety and panic attacks: The conflict model of implosive therapy. *Journal of Integrative and Eclectic Psychotherapy*, 6, 4, 450-461. Levis, D.J. (1988). Observation and experience from clinical practice: A critical ingredient for advancing behavior theory and therapy. *Behavior therapist*, 11, 95-99. Levis, D.J. (1995). Decoding traumatic memory: Implosive theory of psychopathology. In W.

Levis, D.J. (1999). The traumatic memory debate: A failure in scientific communication and

Levis, D.J., & Carrera, R. (1967). Effects of ten hours of implosive therapy in the treatment of out-patients: A preliminary report. *Journal of Abnormal Psychology*, 72, 504-508. Levis, D.J., & Brewer, K.E. (2001). Two-factor fear theory to areas of psychopathology and

Lipschitz, D.S., Kaplan, J.L., Sorkenn, J., Chorney, P., Asnis, G.M. (1996). Childhood abuse, adult assault, and dissociation. *Comprehensive Psychiatry*, 37 (4), 261-266.

memory: An exploratory study. *Journal of Mental Imagery*, 25, 63-78. Knutson, J.F., & Selner, M. (1994). Punitive childhood experiences reported by young adults

over a 10-year period. *Child Abuse & Neglect*, 18 (2), 1155-166.

thesis, Binghamton University, Binghamton, NY, USA.

abuse impacts health status. *Psychometric Medicine*, 58, 4-15.

*in behavior therapy* pp. 49-82). New York: Academic Press.

Washington, D.C.: American Psychological Association.

cooperation. *Applied & Preventive Psychology*, 8, 71-76.

Binghamton, NY, USA.

105, 124-131.

88, 107-120.

manuscript).

to elicit in a non-patient population recall of a reportedly forgotten long-term

gender differences via a multi-modal assessment procedure. Unpublished master's

assessing SAQ-IIIR and gender differences via a multi-modal assessment procedure. Unpublished doctoral dissertation, Binghamton University,

victims: Situational factors and post-traumatic stress. *Journal of Abnormal Psychology*,

Sexual and physical abuse history in gastrointesterology practice: How types of

extension of the conservation of anxiety principle. *Journal of Abnormal Psychology*,

fear/anxiety to psychopathology. In S. Riess & R.R. Bootzin (eds.), *Theoretical issues*

O'Donohue & L. Krasner (eds.). *Theories in behavior therapy* (pp. 173-207).

trauma memory recovery. In R.R. Mowrer & S.B. Klein ( eds.). *Handbook of Contemporary Learning Theories*. New Jersey: Erlbaum Associates, pp. 561-593. Levis, D.J., Rourke, P.A., Wagner, K., Bovier, E., Coleman, S.L., Heron, K., Castelda, B.A.,

and Esch, R.D. (2011). Further evaluation of the Binghamton Abuse Screen (BCAS: An assessment of malingering and the screen's non-face valid status (unpublished

**2**

*USA*

**Sexual Abuse Histories Among**

*Fordham University, Graduate School of Social Service*

**A Descriptive Study**

**Incarcerated Older Adult Offenders:**

Mary Beth Morrissey, Deborah Courtney and Tina Maschi

In the United States today, there are over 2.2 million incarcerated adults held in custody in U.S. state or federal prisons or local jails (Glaze, 2010). Prison facilities are oftentimes filled to capacity or in some cases, so overcrowded that conditions violate the constitutional rights of adults in prison (Brown v. Plata, 2011; Sabol & West, 2009; West & Sabol, 2008). Official statistics paint a contemporary portrait of the 1.5 million adult sentenced prisoners under state or federal jurisdiction who are mostly male (93%) and from diverse racial and ethnic backgrounds including Black American (36%), Caucasian (31%), and Latino (20%)(West & Sabol, 2008). Black males continue to have the highest incarceration rates across all age categories compared to White or Latino males (Sabol & Couture, 2008). There is a growing number of older adults in both state and federal prisons, approaching nearly 5% of inmates 55 and older in custody of state prisons in 2007, and over 7% of inmates 56 and older in federal prison in 2009 (Cox & Lawrence, 2010; Sabol & Couture, 2008). This aging prisoner population, which is five times larger than in 1990, presents a significant public health challenge that the correctional system is not adequately equipped to address (Falter, 1999; Reimer, 2008). Moreover, the high prevalence of trauma among older adults in prison and psychological distress associated with trauma experiences raise serious concerns about the well-being of this population (Krause, 2004). Studies have shown that approximately 93% of juvenile and adult prisoners have had prior exposure to trauma, such as being a victim of and/or witness to sexual abuse (Harlow, 1999). The high prevalence of trauma histories, especially earlier life sexual victimization, within the incarcerated older adult offenders is a major concern. If these traumatic histories go unidentified and untreated, it is likely that unresolved subjective distress about these past events may be heightened, resulting in persistent or resurfacing of post-traumatic stress symptoms or increased likelihood of criminal offending including the perpetration of sexual abuse (Leach, Burgess, Holmwood, 2008). Identifying the types of traumatic experiences, particularly sexual abuse histories, experienced by older adult offenders is important to developing comprehensive approaches to assessment, treatment and program planning for older adults in prison (Dawes, 2009; Rikard & Rosenberg, 2007; Shimkus, 2004). While there are various types of traumatic and stressful experiences in the lives of incarcerated older adults, sexual victimization is an area that demands individualized attention, especially given the high rates of sexual victimization histories prior to prison that occur within this population. For instance, traumatic sexual victimization experiences among offenders occur

**1. Introduction**


*3.3. In Vitro p300 HAT-Mediated Acetylation Reactions*

Acetylation of wt Tau by p300 HAT was performed by combining 500 μL of 86 μM purified wt Tau (dissolved in water), 200 μL of 15 μM p300 HAT, 25 μL of 10 mM acetyl-CoA (Sigma, Saint Louis, MO, USA), and 25 μL of 1 M Tris, pH 8. The acetylation reaction was allowed to proceed for three days at RT (unless stated otherwise).


**4. KYP lemma for** *H*∞ **design problems**

In this section, we show that the *H*∞ design problems given in the previous section are efficiently solved via *semidefinite programming* (Boyd & Vandenberghe, 2004). For this purpose, we first formulate the problems in state-space representation reviewed in Section 2. Then we bring in *Kalman-Yakubovich-Popov* (KYP) lemma (Anderson, 1967; Rantzer, 1996; Tuqan & Vaidyanathan, 1998) to reduce the problems into semidefinite programming.

#### **4.1 State-space representation**

The transfer functions (*P*(*z*) − *Q*(*z*)) *W*(*z*) and (*Q*(*z*)*P*(*z*) − 1) *W*(*z*) in **Problems 1** and **2**, respectively, can be described in a form of

$$T(z) = T\_1(z) + Q(z)T\_2(z),\tag{2}$$

where

6 Digital Signal Processing

*Inverse filtering*, or *deconvolution* is another fundamental issue in signal processing. This problem arises for example in direct-filter design in spline interpolation

Suppose a filter *P*(*z*) is given. Symbolically, the inverse filter of *P*(*z*) is *P*(*z*)−1. However, real

*P*(*z*) = *z* + 0.5

*Q*(*z*) = *P*(*z*)−1 = *z* − 0.5

*P*(*z*) = *z* − 2

*Q*(*z*) = *P*(*z*)−1 = *z* − 0.5

*This has the pole at* |*z*| > 1*, and hence the inverse filter is unstable. On the other hand, suppose*

*P*(*z*) = 1

*Q*(*z*) = *P*(*z*)−1 = *z* − 0.5,

By these examples, the inverse filter *P*(*z*)−1 may unstable or noncausal. Unstable or noncausal filters are difficult to implement in real digital device, and hence we adopt approximation technique; we design an FIR digital filter *Q*(*z*) such that *Q*(*z*)*P*(*z*) ≈ 1. Since FIR filters are always stable and causal, this is a realistic way to design an inverse filter. Our problem is now

**Problem 2** (Inverse filtering)**.** *Given a filter P*(*z*) *which is necessarily not bi-stable or bi-causal (i.e.,*

*Q*(*ejω*)*P*(*ejω*) − 1

*W*(*ejω*) ,

*P*(*z*)−1 *can be unstable or noncausal), find an FIR filter Q*(*z*) *which minimizes*

*ω*∈[0,*π*]

(*QP* − 1)*W*∞ = max

The procedure to solve this problem is shown in Section 4.

*where W is a given stable weighting function.*

*z* − 0.5 ,

*z* − 0.5 ,

*z* − 0.5 .

*z* + 0.5 ,

*z* − 2 .

**3.2 Inverse filtering**

design is not that easy.

*then the inverse is*

*then the inverse is*

*which is noncausal.*

formulated as follows:

(Nagahara & Yamamoto, 2011).

**Example 2.** *Suppose P*(*z*) *is given by*

*Then, the inverse Q*(*z*) := *P*(*z*)−1 *becomes*

*which is stable and causal. Then suppose*

$$T\_1(z) = P(z)W(z), \quad T\_2(z) = -W(z)\_\prime$$

for **Problem 1** and

$$T\_1(z) = -W(z), \quad T\_2(z) = P(z)W(z).$$

for **Problem 2**. Therefore, our problems are described by the following min-max optimization:

$$\min\_{Q(z)\in\mathcal{F}\_N} \|T\_1 + QT\_2\|\_{\infty} = \min\_{Q(z)\in\mathcal{F}\_N} \max\_{\omega\in[0,\pi]} \left| T\_1(e^{j\omega}) + Q(e^{j\omega})T\_2(e^{j\omega}) \right|,\tag{3}$$

where F*N* is the set of *N*-th order FIR filters, that is,

$$\mathcal{F}\_N := \left\{ \mathcal{Q}(z) : \mathcal{Q}(z) = \sum\_{i=0}^N \alpha\_i z^{-i}, \alpha\_i \in \mathbb{R} \right\}. \ .$$

To reduce the problem of minimizing (3) to semidefinite programming, we use state-space representations for *T*1(*z*) and *T*2(*z*) in (2). Let {*Ai*, *Bi*, *Ci*, *Di*} (*i* = 1, 2) are state-space matrices of *Ti*(*z*) in (2), that is,

$$T\_i(z) = \mathbf{C}\_i (zI - A\_i)^{-1} B\_i + D\_i =: \begin{bmatrix} \frac{A\_i \mid B\_i}{\mathbf{C}\_i \mid D\_i} \end{bmatrix} (z), \quad i = 1, 2.1$$

Also, a state-space representation of an FIR filter *Q*(*z*) is given by

$$Q(z) = \sum\_{n=0}^{N} a\_n z^{-n} = \begin{bmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ 0 & 0 & \ddots & \ddots & 0 \\ & 0 & 0 & \ddots & \ddots & 0 \\ \vdots & \vdots & \ddots & 0 & 1 \\ 0 & 0 & \dots & 0 & 0 \\ \hline a\_N a\_{N-1} & \dots & a\_2 & a\_1 \end{bmatrix} (z) =: \begin{bmatrix} A\_q \ \begin{bmatrix} B\_q \end{bmatrix} \end{bmatrix} (z),\tag{4}$$

where *αN*:1 := *αN αN*−1 ... *α*1 .

 *P ejω* − *Q ejω*

*γ*

minimized over the finite frequency range Ωlow = [0, *ω*low].

**5. Finite frequency design of FIR digital filters**

the approximation problem over a finite frequency range.

*V*Ω(*P* − *Q*) := max

can also formulate the inversion problem over a finite frequency range.

*V*Ω(*QP* − 1) := max

Ω ⊂ [0, *π*]*, find an FIR filter Q*(*z*) *which minimizes*

*find an FIR filter Q*(*z*) *which minimizes*

generalized KYP lemma.

Section 7.

*ω*

*ω*low

0 *π*

Fig. 2. Finite frequency approximation (**Problem 3**): the gain of the error *P*(*ejω*) − *Q*(*ejω*) is

Min-Max Design of FIR Digital Filters by Semidefinite Programming 201

where *Mi* is a symmetric matrix and *xi* is the *i*-th entry of *x*. Let *v* ∈ {0, 1}*L* be a vector such that *vx* = *γ*2. Our problem is then described by semidefinite programming as follows:

minimize *vx* subject to *M*(*x*) ≤ 0. By this, we can effectively approach the optimal parameters *α*0, *α*1,..., *αN* by numerical optimization softwares. For MATLAB codes of the semidefinite programming above, see

By the *H*∞ design discussed in the previous section, we can guarantee the maximum gain of the frequency response of *T* = (*P* − *Q*)*W* (approximation) or *T* = (*QP* − 1)*W* (inversion) over the *whole frequency range* [0, *π*]. Some applications, however, do not need minimize the gain over the whole range [0, *π*], but a finite frequency range Ω ⊂ [0, *π*]. Design of noise shaping ∆Σ modulators is one example of such requirement (Nagahara & Yamamoto, 2009). In this section, we consider such optimization, called *finite frequency optimization*. We first consider

**Problem 3** (Finite frequency approximation)**.** *Given a filter P*(*z*) *and a finite frequency range*

Figure 2 illustrates the above problem for a finite frequency range Ω = Ωlow = [0, *ω*low], where *ω*low ∈ (0, *π*]. We seek an FIR filter which minimizes the gain of the error *P*(*ejω*) − *Q*(*ejω*) over the finite frequency range Ω, and do not care about the other range [0, *π*] \ Ω. We

**Problem 4** (Finite frequency inversion)**.** *Given a filter P*(*z*) *and a finite frequency range* Ω ⊂ [0, *π*]*,*

These problems are also fundamental in digital signal processing. We will show in the next section that these problems can be also described in semidefinite programming via

*ω*∈Ω

*P*(*ejω*) − *Q*(*ejω*)

*Q*(*ejω*)*P*(*ejω*) − 1

 .

> .

*ω*∈Ω

By using these state-space matrices, we obtain a state-space representation of *T*(*z*) in (2) as

$$T(z) = \begin{bmatrix} A\_1 & 0 & 0 \\ 0 & A\_2 & 0 \\ 0 & B\_q \mathbf{C\_2} & A\_q \\ \hline \mathbf{C\_1} & \mathfrak{a\_0} \mathbf{C\_2} & \mathfrak{a\_{N:1}} \begin{bmatrix} B\_1 \\ B\_0 D\_2 \end{bmatrix} \\ \end{bmatrix} (z) =: \begin{bmatrix} A & \mid & B \\ \hline \mathbf{C(a\_{N:0})} \big| D(a\_0) \end{bmatrix} (z). \tag{5}$$

Note that the FIR parameters *α*0, *α*1,..., *αN* depend affinely on *C* and *D*, and are independent of *A* and *B*. This property is a key to describe our problem into semidefinite programming.

#### **4.2 Semidefinite programming by KYP lemma**

The optimization in (3) can be equivalently described by the following minimization problem:

$$\begin{aligned} \text{minimize } & \gamma \text{ subject to } Q(z) \in \mathcal{F}\_N \text{ and} \\ & \max\_{\omega \in [0, \pi]} \left| T\_1(e^{j\omega}) + Q(e^{j\omega}) T\_2(e^{j\omega}) \right| \le \gamma. \end{aligned} \tag{6}$$

To describe this optimization in semidefinite programming, we adopt the following lemma (Anderson, 1967; Rantzer, 1996; Tuqan & Vaidyanathan, 1998):

**Lemma 1** (KYP lemma)**.** *Suppose*

$$T(z) = \begin{bmatrix} \frac{A \mid B}{C \mid D} \end{bmatrix} (z)^l$$

*is stable, and the state-space representation* {*A*, *B*, *C*, *D*} *of T*(*z*) *is minimal*1*. Let γ* > 0*. Then the following are equivalent conditions:*

*1. T*∞ ≤ *γ.*

*2. There exists a positive definite matrix X such that*

$$
\begin{bmatrix}
A^\top XA - X & A^\top XB & \mathbf{C}^\top \\
B^\top XA & B^\top XB - \gamma^2 & D \\
\mathbf{C} & D & -1
\end{bmatrix} \le 0.
$$

By using this lemma, we obtain the following theorem:

**Theorem 1.** *The inequality (6) holds if and only if there exists X* > 0 *such that*

$$
\begin{bmatrix}
\boldsymbol{A}^{\top}\boldsymbol{X}\boldsymbol{A} - \boldsymbol{X} & \boldsymbol{A}^{\top}\boldsymbol{X}\boldsymbol{B} & \boldsymbol{C}(\boldsymbol{a}\_{N:0})^{\top} \\
\boldsymbol{B}^{\top}\boldsymbol{X}\boldsymbol{A} & \boldsymbol{B}^{\top}\boldsymbol{X}\boldsymbol{B} - \gamma^{2} & \boldsymbol{D}(\boldsymbol{a}\_{0}) \\
\boldsymbol{C}(\boldsymbol{a}\_{N:0}) & \boldsymbol{D}(\boldsymbol{a}\_{0}) & -1
\end{bmatrix} \leq \boldsymbol{0},\tag{7}
$$

*where A, B, C*(*αN*:0)*, and D*(*α*0) *are given in (5).*

By this, the optimal FIR parameters *α*0, *α*1,..., *αN* can be obtained as follows. Let *x* be the vector consisting of all variables in *αN*:0, *X*, and *γ*2 in (7). The matrix in (7) is *affine* with respect to these variables, and hence, can be rewritten in the form

$$M(\mathfrak{x}) = M\_0 + \sum\_{i=1}^{L} M\_i \mathfrak{x}\_{i\nu}$$

1 For minimality of state-space representation, see Section 2 or Chapter 26 in (Rugh, 1996).

8 Digital Signal Processing

Note that the FIR parameters *α*0, *α*1,..., *αN* depend affinely on *C* and *D*, and are independent of *A* and *B*. This property is a key to describe our problem into semidefinite programming.

The optimization in (3) can be equivalently described by the following minimization problem:

minimize *γ* subject to *Q*(*z*) ∈ F*N* and

*T*1(*ejω*) + *Q*(*ejω*)*T*2(*ejω*)

To describe this optimization in semidefinite programming, we adopt the following lemma

 *A B C D* (*z*)

*is stable, and the state-space representation* {*A*, *B*, *C*, *D*} *of T*(*z*) *is minimal*1*. Let γ* > 0*. Then the*

*AXA* − *X AXB C BXA BXB* − *γ*2 *D C D* −1

*AXA* − *X AXB C*(*αN*:0) *BXA BXB* − *γ*2 *D*(*α*0) *C*(*αN*:0) *D*(*α*0) −1

By this, the optimal FIR parameters *α*0, *α*1,..., *αN* can be obtained as follows. Let *x* be the vector consisting of all variables in *αN*:0, *X*, and *γ*2 in (7). The matrix in (7) is *affine* with

> *L* ∑ *i*=1

*Mixi*,

*M*(*x*) = *M*0 +

1 For minimality of state-space representation, see Section 2 or Chapter 26 in (Rugh, 1996).

*T*(*z*) =

 (*z*) =:

 *A B C*(*αN*:0) *D*(*α*0)

>

> > ≤ 0.

≤ *γ*. (6)

≤ 0, (7)

(*z*). (5)

By using these state-space matrices, we obtain a state-space representation of *T*(*z*) in (2) as

*T*(*z*) =

**Lemma 1** (KYP lemma)**.** *Suppose*

*following are equivalent conditions:*

*2. There exists a positive definite matrix X such that*

*1. T*∞ ≤ *γ.*

**4.2 Semidefinite programming by KYP lemma**

*A*1 0 0 *B*1 0 *A*2 0 *B*2 0 *BqC*2 *Aq BqD*2 *C*1 *α*0*C*2 *αN*:1 *D*1 + *α*0*D*2

> max *ω*∈[0,*π*]

(Anderson, 1967; Rantzer, 1996; Tuqan & Vaidyanathan, 1998):

**Theorem 1.** *The inequality (6) holds if and only if there exists X* > 0 *such that*

respect to these variables, and hence, can be rewritten in the form

By using this lemma, we obtain the following theorem:

*where A, B, C*(*αN*:0)*, and D*(*α*0) *are given in (5).*

Fig. 2. Finite frequency approximation (**Problem 3**): the gain of the error *P*(*ejω*) − *Q*(*ejω*) is minimized over the finite frequency range Ωlow = [0, *ω*low].

where *Mi* is a symmetric matrix and *xi* is the *i*-th entry of *x*. Let *v* ∈ {0, 1}*L* be a vector such that *vx* = *γ*2. Our problem is then described by semidefinite programming as follows:

minimize *vx* subject to *M*(*x*) ≤ 0.

By this, we can effectively approach the optimal parameters *α*0, *α*1,..., *αN* by numerical optimization softwares. For MATLAB codes of the semidefinite programming above, see Section 7.

#### **5. Finite frequency design of FIR digital filters**

By the *H*∞ design discussed in the previous section, we can guarantee the maximum gain of the frequency response of *T* = (*P* − *Q*)*W* (approximation) or *T* = (*QP* − 1)*W* (inversion) over the *whole frequency range* [0, *π*]. Some applications, however, do not need minimize the gain over the whole range [0, *π*], but a finite frequency range Ω ⊂ [0, *π*]. Design of noise shaping ∆Σ modulators is one example of such requirement (Nagahara & Yamamoto, 2009). In this section, we consider such optimization, called *finite frequency optimization*. We first consider the approximation problem over a finite frequency range.

**Problem 3** (Finite frequency approximation)**.** *Given a filter P*(*z*) *and a finite frequency range* Ω ⊂ [0, *π*]*, find an FIR filter Q*(*z*) *which minimizes*

$$V\_{\Omega}(P - Q) := \max\_{\omega \in \Omega} \left| P(e^{j\omega}) - Q(e^{j\omega}) \right|.$$

Figure 2 illustrates the above problem for a finite frequency range Ω = Ωlow = [0, *ω*low], where *ω*low ∈ (0, *π*]. We seek an FIR filter which minimizes the gain of the error *P*(*ejω*) − *Q*(*ejω*) over the finite frequency range Ω, and do not care about the other range [0, *π*] \ Ω. We can also formulate the inversion problem over a finite frequency range.

**Problem 4** (Finite frequency inversion)**.** *Given a filter P*(*z*) *and a finite frequency range* Ω ⊂ [0, *π*]*, find an FIR filter Q*(*z*) *which minimizes*

$$V\_{\Omega}(QP-1) := \max\_{\omega \in \Omega} \left| Q(e^{j\omega})P(e^{j\omega}) - 1 \right|.$$

These problems are also fundamental in digital signal processing. We will show in the next section that these problems can be also described in semidefinite programming via generalized KYP lemma.

*where*

*that*

*M*1(*X*, *Y*) = *AXA* + *YAe*−*jω*0 + *AYejω*0 − *X* − 2*Y* cos *r*,

Min-Max Design of FIR Digital Filters by Semidefinite Programming 203

**Theorem 2.** *The inequality (10) holds if and only if there exist symmetric matrices Y* > 0 *and X such*

By this theorem, we can obtain the coefficients *α*0,..., *αN* of the optimal FIR filter by semidefinite programming as mentioned in Section 4. MATLAB codes for the semidefinite

In this section, we give MATLAB codes for the semidefinite programming derived in previous sections. Note that the MATLAB codes for solving **Problems 1** to **4** are also available at the

http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/fir/ Note also that to execute the codes in this section, Control System Toolbox (Mathworks, 2010), YALMIP (Löfberg, 2004), and SeDuMi (Sturm, 2001) are needed. YALMIP and SeDuMi are free softwares for solving optimization problems including semidefinite programming which

% H-infinity optimal approximated FIR filter Q(z) which minimizes

*M*1(*X*, *Y*) *M*2(*X*, *Y*) *C*(*αN*:0) *M*2(*X*, *Y*) *M*3(*X*, *γ*2) *D*(*α*0) *C*(*αN*:0) *D*(*α*0) −1

*M*3(*X*, *γ*2) = *BXB* − *γ*2, *ω*0 = *ω*1 + *ω*2

*where M*1*, M*2*, and M*3 *are given in (11), A, B, C*(*αN*:0)*, and D*(*α*0) *are given in (9).*

By using this lemma, we obtain the following theorem:

**7. MATLAB codes for semidefinite programming**

**7.1 FIR approximation of IIR filters by** *H*∞ **norm**

function [q,gmin] = approxFIRhinf(P,W,N); % [q,gmin]=approxFIRhinf(P,W) computes the

% the maximum frequency gain of (P-Q)W. % This design uses SDP via the KYP lemma.

% P: Target stable linear system in SS object % W: Weighting stable linear system in SS object % N: Order of the FIR filter to be designed

% q: The optimal FIR filter coefficients

programming are shown in Section 7.

following web site:

is treated in this chapter.

% J(Q) = ||(P-Q)W||,

% gmin: The optimal value

%

%

%

% Inputs:

% Outputs:

*M*2(*X*, *Y*) = *AXB* + *YBe*−*jω*0 , *M*2(*X*, *Y*) = *AXB* + *YBejω*0 ,

2 , *r* = *ω*2 − *ω*1

 ≤ 0, 2 .

(11)

#### **6. Generalized KYP lemma for finite frequency design problems**

In this section, we reduce the problems given in the previous section to semidefinite programming. As in the *H*∞ optimization, we first formulate the problems in state-space representation, and then derive semidefinite programming via *generalized KYP lemma* (Iwasaki & Hara, 2005).

#### **6.1 State-space representation**

As in the *H*∞ optimization in Section 4, we employ state-space representation. Let *T*(*z*) = *P*(*z*) − *Q*(*z*) for the approximation problem or *T*(*z*) = *P*(*z*)*Q*(*z*) − 1 for the inversion problem. Then *T*(*z*) can be described by *T*(*z*) = *T*1(*z*) + *Q*(*z*)*T*2(*z*) as in (2). Then our problems are described by the following min-max optimization:

$$\min\_{Q(z)\in\mathcal{F}\_N} V\_\Omega \left( T\_1 + Q T\_2 \right) = \min\_{Q(z)\in\mathcal{F}\_N} \max\_{\omega\in\Omega} \left| T\_1(e^{j\omega}) + Q(e^{j\omega}) T\_2(e^{j\omega}) \right|. \tag{8}$$

Let {*Ai*, *Bi*, *Ci*, *Di*}, *i* = 1, 2 be state-space matrices of *Ti*(*z*). By using the same technique as in Section 4, we can obtain a state-space representation of *T*(*z*) as

$$T(z) = \begin{bmatrix} A & B \\ \hline \mathbb{C}(\mathfrak{a}\_{N:0}) \mathop{\mid} D(\mathfrak{a}\_0) \end{bmatrix} (z), \tag{9}$$

where *αN*:0 = [*αN*,..., *α*0] is the coefficient vector of the FIR filter to be designed as defined in (4).

#### **6.2 Semidefinite programming by generalized KYP lemma**

The optimization in (8) can be equivalently described by the following problem:

$$\text{minimize } \gamma \text{ subject to } Q(z) \in \mathcal{F}\_N \text{ and }$$

$$\max\_{\omega \in \Omega} \left| T\_1(e^{j\omega}) + Q(e^{j\omega})T\_2(e^{j\omega}) \right| \le \gamma \tag{10}$$

To describe this optimization in semidefinite programming, we adopt the following lemma (Iwasaki & Hara, 2005):

**Lemma 2** (Generalized KYP Lemma)**.** *Suppose*

$$T(z) = \begin{bmatrix} A \mid B \\ \hline C \mid D \end{bmatrix} (z)$$

*is stable, and the state-space representation* {*A*, *B*, *C*, *D*} *of T*(*z*) *is minimal. Let* Ω *be a closed interval* [*ω*1, *ω*2] ⊂ [0, *π*]*. Let γ* > 0*. Then the following are equivalent conditions:*

$$
\begin{bmatrix}
\begin{matrix}
M\_1(X,Y) & M\_2(X,Y) & \mathbb{C}^\top \\
\overline{M\_2}(X,Y)^\top & M\_3(X,\gamma^2) & D \\
\mathbb{C} & D & -1
\end{matrix} \end{bmatrix} \le 0\_\prime
$$

*where*

10 Digital Signal Processing

In this section, we reduce the problems given in the previous section to semidefinite programming. As in the *H*∞ optimization, we first formulate the problems in state-space representation, and then derive semidefinite programming via *generalized KYP lemma*

As in the *H*∞ optimization in Section 4, we employ state-space representation. Let *T*(*z*) = *P*(*z*) − *Q*(*z*) for the approximation problem or *T*(*z*) = *P*(*z*)*Q*(*z*) − 1 for the inversion problem. Then *T*(*z*) can be described by *T*(*z*) = *T*1(*z*) + *Q*(*z*)*T*2(*z*) as in (2). Then our problems are

> max *ω*∈Ω

> ≤ 0,

*T*1(*ejω*) + *Q*(*ejω*)*T*2(*ejω*)

(*z*), (9)

≤ *γ* (10)

. (8)

*Q*(*z*)∈F*N*

Let {*Ai*, *Bi*, *Ci*, *Di*}, *i* = 1, 2 be state-space matrices of *Ti*(*z*). By using the same technique as in

 *A B C*(*αN*:0) *D*(*α*0)

where *αN*:0 = [*αN*,..., *α*0] is the coefficient vector of the FIR filter to be designed as defined in

minimize *γ* subject to *Q*(*z*) ∈ F*N* and

*T*1(*ejω*) + *Q*(*ejω*)*T*2(*ejω*)

To describe this optimization in semidefinite programming, we adopt the following lemma

 *A B C D* (*z*)

*is stable, and the state-space representation* {*A*, *B*, *C*, *D*} *of T*(*z*) *is minimal. Let* Ω *be a closed interval*

*M*1(*X*,*Y*) *M*2(*X*, *Y*) *C M*2(*X*, *Y*) *M*3(*X*, *γ*2) *D C D* −1

**6. Generalized KYP lemma for finite frequency design problems**

*V*Ω (*T*1 + *QT*2) = min

*T*(*z*) =

The optimization in (8) can be equivalently described by the following problem:

*T*(*z*) =

[*ω*1, *ω*2] ⊂ [0, *π*]*. Let γ* > 0*. Then the following are equivalent conditions:*

Section 4, we can obtain a state-space representation of *T*(*z*) as

**6.2 Semidefinite programming by generalized KYP lemma**

max *ω*∈Ω

**Lemma 2** (Generalized KYP Lemma)**.** *Suppose*

 *T*(*ejω*) ≤ *γ.*

*2. There exist symmetric matrices Y* > 0 *and X such that*

(Iwasaki & Hara, 2005).

(Iwasaki & Hara, 2005):

*1. V*Ω(*T*) = max*ω*∈[*ω*1,*ω*2]

(4).

**6.1 State-space representation**

min *Q*(*z*)∈F*N*

described by the following min-max optimization:

$$\begin{aligned} M\_1(X,Y) &= A^\top XA + YA e^{-j\omega\_0} + A^\top Y e^{j\omega\_0} - X - 2Y \cos r, \\ M\_2(X,Y) &= A^\top XB + YB e^{-j\omega\_0}, \quad \overline{M}\_2(X,Y) = A^\top XB + YB e^{j\omega\_0}, \\ M\_3(X,\gamma^2) &= B^\top XB - \gamma^2, \quad \omega\_0 = \frac{\omega\_1 + \omega\_2}{2}, \quad r = \frac{\omega\_2 - \omega\_1}{2}. \end{aligned} \tag{11}$$

By using this lemma, we obtain the following theorem:

**Theorem 2.** *The inequality (10) holds if and only if there exist symmetric matrices Y* > 0 *and X such that*

$$
\begin{bmatrix}
\overline{M\_1}(X,Y) & M\_2(X,Y) & C(\mathfrak{a}\_{N:0})^\top \\
\overline{M\_2}(X,Y)^\top & M\_3(X,\gamma^2) & D(\mathfrak{a}\_0) \\
\overline{C(\mathfrak{a}\_{N:0})} & D(\mathfrak{a}\_0) & -1
\end{bmatrix} \le 0,
$$

*where M*1*, M*2*, and M*3 *are given in (11), A, B, C*(*αN*:0)*, and D*(*α*0) *are given in (9).*

By this theorem, we can obtain the coefficients *α*0,..., *αN* of the optimal FIR filter by semidefinite programming as mentioned in Section 4. MATLAB codes for the semidefinite programming are shown in Section 7.

#### **7. MATLAB codes for semidefinite programming**

In this section, we give MATLAB codes for the semidefinite programming derived in previous sections. Note that the MATLAB codes for solving **Problems 1** to **4** are also available at the following web site:

```
http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/fir/
```
Note also that to execute the codes in this section, Control System Toolbox (Mathworks, 2010), YALMIP (Löfberg, 2004), and SeDuMi (Sturm, 2001) are needed. YALMIP and SeDuMi are free softwares for solving optimization problems including semidefinite programming which is treated in this chapter.

#### **7.1 FIR approximation of IIR filters by** *H*∞ **norm**

```
function [q,gmin] = approxFIRhinf(P,W,N);
% [q,gmin]=approxFIRhinf(P,W) computes the
% H-infinity optimal approximated FIR filter Q(z) which minimizes
% J(Q) = ||(P-Q)W||,
% the maximum frequency gain of (P-Q)W.
% This design uses SDP via the KYP lemma.
%
% Inputs:
% P: Target stable linear system in SS object
% W: Weighting stable linear system in SS object
% N: Order of the FIR filter to be designed
%
% Outputs:
% q: The optimal FIR filter coefficients
% gmin: The optimal value
%
```
% H-infinity optimal (delayed) inverse FIR filter Q(z) which minimizes

Min-Max Design of FIR Digital Filters by Semidefinite Programming 205

% J(Q) = ||(QP-z^(-n))W||,

% gmin: The optimal value

[A1,B1,C1,D1]=ssdata(T1); [A2,B2,C2,D2]=ssdata(T2);

%% FIR filter to be designed Aq = circshift(eye(N),-1);

%% Semidefinite Programming

A = [A1, zeros(n1,n2), zeros(n1,N); zeros(n2,n1), A2, zeros(n2,N);

zeros(N,n1),Bq\*C2, Aq];

X = sdpvar(NN,NN,'symmetric');

alpha\_N1 = sdpvar(1,N); alpha\_0 = sdpvar(1,1); gamma = sdpvar(1,1);

Bq = [zeros(N-1,1);1];

B = [B1;B2;Bq\*D2];

NN = size(A,1);

%

%

%

end

% Inputs:

% Outputs:

if nargin==3 n=0

T2 = P\*W;

%% Initialization z = tf('z'); T1 = -z^(-n)\*W;

n1 = size(A1,1); n2 = size(A2,1);

Aq(N,1) = 0;

% the maximum frequency gain of (QP-z^(-n))W. % This design uses SDP via the KYP lemma.

% P: Target stable linear system in SS object % W: Weighting stable linear system in SS object % N: Order of the FIR filter to be designed

% q: The optimal FIR filter coefficients

% n: Delay (this can be omitted; default value=0);

```
%% Initialization
T1 = P*W;
T2 = -W;
[A1,B1,C1,D1]=ssdata(T1);
[A2,B2,C2,D2]=ssdata(T2);
n1 = size(A1,1);
n2 = size(A2,1);
%% FIR filter to be designed
Aq = circshift(eye(N),-1);
Aq(N,1) = 0;
Bq = [zeros(N-1,1);1];
%% Semidefinite Programming
A = [A1, zeros(n1,n2), zeros(n1,N);
     zeros(n2,n1), A2, zeros(n2,N);
     zeros(N,n1),Bq*C2, Aq];
B = [B1;B2;Bq*D2];
NN = size(A,1);
X = sdpvar(NN,NN,'symmetric');
alpha_N1 = sdpvar(1,N);
alpha_0 = sdpvar(1,1);
gamma = sdpvar(1,1);
M1 = A'*X*A-X;
M2 = A'*X*B;
M3 = B'*X*B-gamma;
C = [C1, alpha_0*C2, alpha_N1];
D = D1 + alpha_0*D2;
M = [M1, M2, C'; M2', M3, D; C, D, -gamma];
F = set(M < 0) + set(X > 0) + set(gamma > 0);
solvesdp(F,gamma);
%% Optimal FIR filter coefficients
q = fliplr([double(alpha_N1),double(alpha_0)]);
gmin = double(gamma);
```
#### **7.2 Inverse FIR filtering by** *H*∞ **norm**

```
function [q,gmin] = inverseFIRhinf(P,W,N,n);
% [q,gmin]=inverseFIRhinf(P,W,N,n) computes the
```
12 Digital Signal Processing

%% Initialization

n1 = size(A1,1); n2 = size(A2,1);

Aq(N,1) = 0;

[A1,B1,C1,D1]=ssdata(T1); [A2,B2,C2,D2]=ssdata(T2);

%% FIR filter to be designed Aq = circshift(eye(N),-1);

%% Semidefinite Programming

A = [A1, zeros(n1,n2), zeros(n1,N); zeros(n2,n1), A2, zeros(n2,N);

zeros(N,n1),Bq\*C2, Aq];

X = sdpvar(NN,NN,'symmetric');

C = [C1, alpha\_0\*C2, alpha\_N1];

%% Optimal FIR filter coefficients

M = [M1, M2, C'; M2', M3, D; C, D, -gamma];

F = set(M < 0) + set(X > 0) + set(gamma > 0);

q = fliplr([double(alpha\_N1),double(alpha\_0)]);

function [q,gmin] = inverseFIRhinf(P,W,N,n); % [q,gmin]=inverseFIRhinf(P,W,N,n) computes the

alpha\_N1 = sdpvar(1,N); alpha\_0 = sdpvar(1,1); gamma = sdpvar(1,1);

Bq = [zeros(N-1,1);1];

B = [B1;B2;Bq\*D2];

NN = size(A,1);

M1 = A'\*X\*A-X; M2 = A'\*X\*B;

M3 = B'\*X\*B-gamma;

D = D1 + alpha\_0\*D2;

solvesdp(F,gamma);

gmin = double(gamma);

**7.2 Inverse FIR filtering by** *H*∞ **norm**

T1 = P\*W; T2 = -W;

```
% H-infinity optimal (delayed) inverse FIR filter Q(z) which minimizes
% J(Q) = ||(QP-z^(-n))W||,
% the maximum frequency gain of (QP-z^(-n))W.
% This design uses SDP via the KYP lemma.
%
% Inputs:
% P: Target stable linear system in SS object
% W: Weighting stable linear system in SS object
% N: Order of the FIR filter to be designed
% n: Delay (this can be omitted; default value=0);
%
% Outputs:
% q: The optimal FIR filter coefficients
% gmin: The optimal value
%
if nargin==3
    n=0
end
%% Initialization
z = tf('z');
T1 = -z^(-n)*W;
T2 = P*W;
[A1,B1,C1,D1]=ssdata(T1);
[A2,B2,C2,D2]=ssdata(T2);
n1 = size(A1,1);
n2 = size(A2,1);
%% FIR filter to be designed
Aq = circshift(eye(N),-1);
Aq(N,1) = 0;
Bq = [zeros(N-1,1);1];
%% Semidefinite Programming
A = [A1, zeros(n1,n2), zeros(n1,N);
     zeros(n2,n1), A2, zeros(n2,N);
     zeros(N,n1),Bq*C2, Aq];
B = [B1;B2;Bq*D2];
NN = size(A,1);
X = sdpvar(NN,NN,'symmetric');
alpha_N1 = sdpvar(1,N);
alpha_0 = sdpvar(1,1);
gamma = sdpvar(1,1);
```
B = [B1;-Bq];

NN = size(A,1);

g = sdpvar(1,1);

M3r = B'\*P\*B-g;

solvesdp(F,g);

gmin = double(g);

%

% Inputs:

C = [C1, alpha\_N1]; D = D1 - alpha\_0;

omega0 = (Omega(1)+Omega(2))/2; omegab = (Omega(2)-Omega(1))/2;

P = sdpvar(NN,NN,'symmetric'); Q = sdpvar(NN,NN,'symmetric');

M2r = A'\*P\*B+Q\*B\*cos(omega0);

M21i = -Q\*B\*sin(omega0); M22i = B'\*Q\*sin(omega0);

M = [Mr, Mi; -Mi, Mr];

M1i = A'\*Q\*sin(omega0)-Q\*A\*sin(omega0);

F = set(M < 0) + set(Q > 0) + set(g > 0);

q = fliplr([double(alpha\_N1),double(alpha\_0)]);

function [q,gmin] = inverseFIRff(P,Omega,N,n); % [q,gmin]=inverseFIRff(P,Omega,N,n) computes the

% P: Target stable linear system in SS object % Omega: Frequency band in 1x2 vector [w1,w2] % N: Order of the FIR filter to be designed

Mr = [M1r,M2r,C';M2r',M3r,D;C,D,-1];

%% Optimal FIR filter coefficients

**7.4 Inverse FIR filtering by finite-frequency min-max**

M1r = A'\*P\*A+Q\*A\*cos(omega0)+A'\*Q\*cos(omega0)-P-2\*Q\*cos(omegab);

Min-Max Design of FIR Digital Filters by Semidefinite Programming 207

% Finite-frequency optimal (delayed) inverse FIR filter Q(z) which minimizes

Mi = [M1i, M21i, zeros(NN,1);M22i, 0, 0; zeros(1,NN),0,0];

% J(Q) = max{|Q(exp(jw)P(exp(jw))-exp(-jwn)|, w in Omega}.

% This design uses SDP via the generalized KYP lemma.

% n: Delay (this can be omitted; default value=0);

% the maximum frequency gain of QP-z^(-n) in a frequency band Omega.

alpha\_N1 = sdpvar(1,N); alpha\_0 = sdpvar(1,1);

```
M1 = A'*X*A-X;
M2 = A'*X*B;
M3 = B'*X*B-gamma;
C = [C1, alpha_0*C2, alpha_N1];
D = D1 + alpha_0*D2;
M = [M1, M2, C'; M2', M3, D; C, D, -gamma];
F = set(M < 0) + set(X > 0) + set(gamma > 0);
solvesdp(F,gamma);
%% Optimal FIR filter coefficients
q = fliplr([double(alpha_N1),double(alpha_0)]);
gmin = double(gamma);
```
#### **7.3 FIR approximation of IIR filters by finite-frequency min-max**

```
function [q,gmin] = approxFIRff(P,Omega,N);
% [q,gmin]=approxFIRff(P,Omega,N) computes the
% Finite-frequency optimal approximated FIR filter Q(z)
  which minimizes
% J(Q) = max{|P(exp(jw))-Q(exp(jw))|, w in Omega}l.
% the maximum frequency gain of P-Q in a frequency band Omega.
% This design uses SDP via the generalized KYP lemma.
%
% Inputs:
% P: Target stable linear system in SS object
% Omega: Frequency band in 1x2 vector [w1,w2]
% N: Order of the FIR filter to be designed
%
% Outputs:
% q: The optimal FIR filter coefficients
% gmin: The optimal value
%
%% Initialization
[A1,B1,C1,D1]=ssdata(P);
n1 = size(A1,1);
%% FIR filter to be designed
Aq = circshift(eye(N),-1);
Aq(N,1) = 0;
Bq = [zeros(N-1,1);1];
%% Semidefinite Programming
A = blkdiag(A1,Aq);
```
14 Digital Signal Processing

M1 = A'\*X\*A-X; M2 = A'\*X\*B;

M3 = B'\*X\*B-gamma;

D = D1 + alpha\_0\*D2;

solvesdp(F,gamma);

gmin = double(gamma);

which minimizes

%

%

%

% Inputs:

% Outputs:

%% Initialization

n1 = size(A1,1);

Aq(N,1) = 0;

C = [C1, alpha\_0\*C2, alpha\_N1];

%% Optimal FIR filter coefficients

M = [M1, M2, C'; M2', M3, D; C, D, -gamma];

F = set(M < 0) + set(X > 0) + set(gamma > 0);

q = fliplr([double(alpha\_N1),double(alpha\_0)]);

**7.3 FIR approximation of IIR filters by finite-frequency min-max** function [q,gmin] = approxFIRff(P,Omega,N); % [q,gmin]=approxFIRff(P,Omega,N) computes the

% P: Target stable linear system in SS object % Omega: Frequency band in 1x2 vector [w1,w2] % N: Order of the FIR filter to be designed

% q: The optimal FIR filter coefficients

% gmin: The optimal value

[A1,B1,C1,D1]=ssdata(P);

Bq = [zeros(N-1,1);1];

A = blkdiag(A1,Aq);

%% FIR filter to be designed Aq = circshift(eye(N),-1);

%% Semidefinite Programming

% Finite-frequency optimal approximated FIR filter Q(z)

% the maximum frequency gain of P-Q in a frequency band Omega.

% J(Q) = max{|P(exp(jw))-Q(exp(jw))|, w in Omega}l.

% This design uses SDP via the generalized KYP lemma.

```
B = [B1;-Bq];
NN = size(A,1);
omega0 = (Omega(1)+Omega(2))/2;
omegab = (Omega(2)-Omega(1))/2;
P = sdpvar(NN,NN,'symmetric');
Q = sdpvar(NN,NN,'symmetric');
alpha_N1 = sdpvar(1,N);
alpha_0 = sdpvar(1,1);
g = sdpvar(1,1);
C = [C1, alpha_N1];
D = D1 - alpha_0;
M1r = A'*P*A+Q*A*cos(omega0)+A'*Q*cos(omega0)-P-2*Q*cos(omegab);
M2r = A'*P*B+Q*B*cos(omega0);
M3r = B'*P*B-g;
M1i = A'*Q*sin(omega0)-Q*A*sin(omega0);
M21i = -Q*B*sin(omega0);
M22i = B'*Q*sin(omega0);
Mr = [M1r,M2r,C';M2r',M3r,D;C,D,-1];
Mi = [M1i, M21i, zeros(NN,1);M22i, 0, 0; zeros(1,NN),0,0];
M = [Mr, Mi; -Mi, Mr];
F = set(M < 0) + set(Q > 0) + set(g > 0);
solvesdp(F,g);
%% Optimal FIR filter coefficients
q = fliplr([double(alpha_N1),double(alpha_0)]);
gmin = double(g);
7.4 Inverse FIR filtering by finite-frequency min-max
function [q,gmin] = inverseFIRff(P,Omega,N,n);
% [q,gmin]=inverseFIRff(P,Omega,N,n) computes the
% Finite-frequency optimal (delayed) inverse FIR filter Q(z) which minimizes
% J(Q) = max{|Q(exp(jw)P(exp(jw))-exp(-jwn)|, w in Omega}.
% the maximum frequency gain of QP-z^(-n) in a frequency band Omega.
% This design uses SDP via the generalized KYP lemma.
%
% Inputs:
% P: Target stable linear system in SS object
% Omega: Frequency band in 1x2 vector [w1,w2]
% N: Order of the FIR filter to be designed
```

```
% n: Delay (this can be omitted; default value=0);
```
M1i = A'\*Q\*sin(omega0)-Q\*A\*sin(omega0);

F = set(M < 0) + set(Q > 0) + set(g > 0);

q = fliplr([double(alpha\_N1),double(alpha\_0)]);

in the frequency band [0, *π*/2], *Q*1(*z*) shows the larger error than *Q*2(*z*).

Mi = [M1i, M21i, zeros(NN,1);M22i, 0, 0; zeros(1,NN),0,0];

Min-Max Design of FIR Digital Filters by Semidefinite Programming 209

By the MATLAB codes given in the previous section, we design FIR filters for **Problems 1** and **3**. Let the FIR filter order *N* = 8. The target filter is the second order lowpass Butterworth filter with cutoff frequency *π*/2. This can be computed by butter(2,1/2) in MATLAB. The weighting transfer function in **Problem 1** is chosen by an 8th-order lowpass Chebyshev filter, computed by cheby1(8,1/2,1/2) in MATLAB. The frequency band for **Problem 3** is Ω = [0, *π*/2]. Figure 3 shows the gain of the error *E*(*z*) := *P*(*z*) − *Q*(*z*). We can see that the *H*∞ optimal filter (the solution of **Problem 1**), say *Q*1(*z*), shows the lower *H*∞ norm than the finite-frequency min-max design (the solution of **Problem 3**), say *Q*2(*z*). On the other hand,

0 0.5 1 1.5 2 2.5 3 −110

Fig. 3. The gain of the error *E*(*z*) = *P*(*z*) − *Q*(*z*) for *H*∞ optimization (solid) and

Frequency (rad/sec)

π/2

H∞ norm = −34.2 (dB)

H∞ norm = −22.5 (dB)

Mr = [M1r,M2r,C';M2r',M3r,D;C,D,-1];

%% Optimal FIR filter coefficients

−100 −90 −80 −70 −60 −50 −40 −30 −20

−72.4 (dB)

−85.6 (dB)

finite-frequency min-max optimization (dash)

Error (dB)

M21i = -Q\*B\*sin(omega0); M22i = B'\*Q\*sin(omega0);

M = [Mr, Mi; -Mi, Mr];

solvesdp(F,g);

gmin = double(g);

**8. Examples**

```
%
% Outputs:
% q: The optimal FIR filter coefficients
% gmin: The optimal value
%
if nargin==3
    n=0
end
%% Initialization
z = tf('z');
T1 = -z^(-n);
T2 = P;
[A1,B1,C1,D1]=ssdata(T1);
[A2,B2,C2,D2]=ssdata(T2);
n1 = size(A1,1);
n2 = size(A2,1);
%% FIR filter to be designed
Aq = circshift(eye(N),-1);
Aq(N,1) = 0;
Bq = [zeros(N-1,1);1];
%% Semidefinite Programming
A = [A1, zeros(n1,n2), zeros(n1,N);
     zeros(n2,n1), A2, zeros(n2,N);
     zeros(N,n1),Bq*C2, Aq];
B = [B1;B2;Bq*D2];
NN = size(A,1);
omega0 = (Omega(1)+Omega(2))/2;
omegab = (Omega(2)-Omega(1))/2;
P = sdpvar(NN,NN,'symmetric');
Q = sdpvar(NN,NN,'symmetric');
alpha_N1 = sdpvar(1,N);
alpha_0 = sdpvar(1,1);
g = sdpvar(1,1);
C = [C1, alpha_0*C2, alpha_N1];
D = D1 + alpha_0*D2;
M1r = A'*P*A+Q*A*cos(omega0)+A'*Q*cos(omega0)-P-2*Q*cos(omegab);
M2r = A'*P*B+Q*B*cos(omega0);
M3r = B'*P*B-g;
```

```
M1i = A'*Q*sin(omega0)-Q*A*sin(omega0);
M21i = -Q*B*sin(omega0);
M22i = B'*Q*sin(omega0);
Mr = [M1r,M2r,C';M2r',M3r,D;C,D,-1];
Mi = [M1i, M21i, zeros(NN,1);M22i, 0, 0; zeros(1,NN),0,0];
M = [Mr, Mi; -Mi, Mr];
F = set(M < 0) + set(Q > 0) + set(g > 0);
solvesdp(F,g);
%% Optimal FIR filter coefficients
q = fliplr([double(alpha_N1),double(alpha_0)]);
gmin = double(g);
```
#### **8. Examples**

16 Digital Signal Processing

% q: The optimal FIR filter coefficients

% gmin: The optimal value

[A1,B1,C1,D1]=ssdata(T1); [A2,B2,C2,D2]=ssdata(T2);

%% FIR filter to be designed Aq = circshift(eye(N),-1);

%% Semidefinite Programming

A = [A1, zeros(n1,n2), zeros(n1,N); zeros(n2,n1), A2, zeros(n2,N);

zeros(N,n1),Bq\*C2, Aq];

omega0 = (Omega(1)+Omega(2))/2; omegab = (Omega(2)-Omega(1))/2;

P = sdpvar(NN,NN,'symmetric'); Q = sdpvar(NN,NN,'symmetric');

C = [C1, alpha\_0\*C2, alpha\_N1];

M2r = A'\*P\*B+Q\*B\*cos(omega0);

M1r = A'\*P\*A+Q\*A\*cos(omega0)+A'\*Q\*cos(omega0)-P-2\*Q\*cos(omegab);

alpha\_N1 = sdpvar(1,N); alpha\_0 = sdpvar(1,1);

Bq = [zeros(N-1,1);1];

B = [B1;B2;Bq\*D2];

NN = size(A,1);

g = sdpvar(1,1);

M3r = B'\*P\*B-g;

D = D1 + alpha\_0\*D2;

%

%

end

T2 = P;

% Outputs:

if nargin==3 n=0

%% Initialization z = tf('z'); T1 = -z^(-n);

n1 = size(A1,1); n2 = size(A2,1);

Aq(N,1) = 0;

By the MATLAB codes given in the previous section, we design FIR filters for **Problems 1** and **3**. Let the FIR filter order *N* = 8. The target filter is the second order lowpass Butterworth filter with cutoff frequency *π*/2. This can be computed by butter(2,1/2) in MATLAB. The weighting transfer function in **Problem 1** is chosen by an 8th-order lowpass Chebyshev filter, computed by cheby1(8,1/2,1/2) in MATLAB. The frequency band for **Problem 3** is Ω = [0, *π*/2]. Figure 3 shows the gain of the error *E*(*z*) := *P*(*z*) − *Q*(*z*). We can see that the *H*∞ optimal filter (the solution of **Problem 1**), say *Q*1(*z*), shows the lower *H*∞ norm than the finite-frequency min-max design (the solution of **Problem 3**), say *Q*2(*z*). On the other hand, in the frequency band [0, *π*/2], *Q*1(*z*) shows the larger error than *Q*2(*z*).

Fig. 3. The gain of the error *E*(*z*) = *P*(*z*) − *Q*(*z*) for *H*∞ optimization (solid) and finite-frequency min-max optimization (dash)

**11**

*Japan*

Baba Tatsuro

*Toshiba Medical Systems Corporation*

**Complex Digital Filter Designs for Audio**

**Processing in Doppler Ultrasound System**

A medical Doppler ultrasound system has a spectrum display that indicates the blood flow direction, whether the blood flows forward or away from a probe. It also has Doppler audio outputs. In particular, the latter is a special process peculiar to the Doppler ultrasound system and separates the blood flow direction and outputs from the left and right speakers. Owing to this function, the existence of a blood flow is quickly detectable. When changing conventional analog signal-processing into digital signal-processing, we researched many processing systems of Doppler audio. First, target performances, such as a response time and direction separation, were set up, and six kinds of digital signal-processing systems were examined. Further, we investigated some new anti-aliasing processing systems unique to Doppler ultrasound system. We compared three kinds of anti-aliasing processing systems. Consequently, we clarified that a complex IIR (infinite impulse response) filter

**2. Outline of Doppler ultrasound system and conventional analog signal-**

Recently, the diagnostic ultrasound system has been popular in many diagnostic fields, such as cardiac, abdomen, and so on. In Section 2.1, an example of diagnostic image and its principle are introduced. In Section 2.2, the phase shift system that is an example of

An example of diagnostic image of a carotid artery is shown in Fig. 1. The upper is a tomogram image and bottom is a spectrum Doppler image. This image expresses the time change of the flow velocity in the PWD (Pulse Wave Doppler) range gate set up in the central of a blood vessel in a tomogram. A horizontal axis and a vertical axis are the flow

Signal processing of the ultrasound echo signal is shown in Fig. 2. An ultrasonic wave is transmitted for every cycle of PRF (pulse repetition frequency: *fs*) in the transceiver processing part of Fig. 2(a), and a reflective echo is received. An ultrasonic beam is scanned in the transverse direction, and envelope detection of the received signal is carried out in the

system has an excellent response and a low calculation load.

**2.1 Outline of Doppler ultrasound system**

representation of conventional analog signal-processing is introduced.

velocities corresponding to Doppler shift frequency and time, respectively.

range direction. This scanning constitutes the tomogram image.

**1. Introduction**

**processing**

#### **9. Conclusion**

In this chapter, we consider four problems, FIR approximation and inverse FIR filtering of FIR/IIR filters by *H*∞ and finite-frequency min-max, which are fundamental in signal processing. By using the KYP and generalized KYP lemmas, the problems are all solvable via semidefinite programming. We show MATLAB codes for the programming, and show examples of designing FIR filters.

#### **10. References**

Mathworks (2010). *Control System Toolbox Users Guide*. URL: *http://www.mathworks.com/products/control/*