# Digital Architectures for UWB Beamforming Using 2D IIR Spatio-Temporal Frequency-Planar Filters

Authors: Soumya Kondapalli; Arjuna Madanayake; Len Bruton
Journal: International Journal of Antennas and Propagation
 (2012)

---

## Abstract

A design method and an FPGA-based prototype implementation of massively parallel systolic-array VLSI architectures
for 2nd-order and 3rd-order frequency-planar beam plane-wave filters are proposed. Frequency-planar beamforming enables highly-directional UWB RF beams at low computational complexity compared to digital phased-array feed techniques. The array factors of the proposed realizations are simulated
and both high-directional selectivity and UWB performance are demonstrated. The proposed architectures operate using 2's complement finite precision digital arithmetic. The real-time throughput is maximized using look-ahead optimization applied locally to each processor in the proposed massively-parallel realization of the filter. From sensitivity theory, it is shown that 15 and 19-bit precision for filter coefficients results in better than 3% error for 2nd- and 3rd-order beam filters. Folding together with Ktimes multiplexing is applied to the proposed beam architectures such that throughput can be traded forK-fold lower complexity for realizing the 2-D fan filter banks. Prototype FPGA circuit implementations of these filters are proposed using a Virtex 6 xc6vsx475t-2ff1759 device. The FPGA-prototyped architectures are evaluated using area (A), critical path delay (T), and metrics AT and AT2. The L2 error energy is used as a metric for evaluating fixed-point noise levels and the accuracy of the finite precision digital arithmetic circuits.


---

## Body

## 1. Introduction

Radio-frequency (RF) two-dimensional (2D) infinite impulse response (IIR) space-time (ST) plane-wave frequency-planar beam filters [1] have potential applications in ultra-wideband (UWB) directional filtering of propagating electromagnetic far-field plane-waves. Such plane-wave filters achieve highly directional beamforming for aperture array applications. The proposed beam filters are designed using the concept of frequency-planar resonant 2D inductor-capacitor (LC) ladder network prototypes having resistive terminations [2]. For example, UWB beam filters can be employed in radar [3], wireless communications [4], radio astronomy [5], and electromagnetic imaging, and sensing [6]. Furthermore, new applications have been proposed in cognitive radio towards enhanced access to radio spectrum (EARS) [7] which requires sensitive spectrum sensing in both space and time domains [8], in turn leading to a strong need for low-complexity directional filters capable of real-time RF operation [9].High attenuation in the stop-band region as well as a sharp transition from filter passband to stop-band is greatly desired in high-performance beamforming systems because such characteristics are important for achieving a better approximation to the ideally “brick-wall” type transition between main beam and stop-band null of the array pattern. The sharpness of the transition from passband (main beam) to stop-band (null region) of the aperture array factor directly depends on the order of the transfer function of the spatio-temporal filter that is employed for UWB beamforming. The primary objective of this work is to explore the real-time hardware architectures that are necessary for realizing frequency-planar beam plane-wave filters corresponding to 2nd- and 3rd-order 2D passive LC ladder low-pass networks. The proposed architectures are an extension of the elementary 1st-order hardware architecture described in [10]. Here, we propose novel massively parallel digital architectures with detailed design equations and complexity studies for beamforming networks based on LC-ladder prototypes of order 2 and 3 having significantly better UWB directionality compared to available 1st-order realizations [10].The 2D frequency-planar beam filters are both practical bounded-input and bounded-output (practical-BIBO) stable [11] and structurally stable under zero initial conditions (ZICs) and can be designed for low computational complexity. We propose massively parallel systolic-array VLSI architectures containing identical and locally interconnected parallel processing core modules (PPCMs) for 2nd-order and 3rd-order beam filters. The proposed architectures are fine-grain pipelined using both inter-PPCM and intra-PPCM registers in order to reduce the critical path delays (CPDs) for achieving maximum clock frequency and temporal bandwidth. The main reason to pipeline the filters is to achieve higher throughputs required for real-time filtering of 2D RF signals derived from time synchronously sampled UWB uniform linear arrays (ULAs) of antenna elements. The proposed systolic-array architectures achieve real-time RF plane-wave filtering at a throughput of one-frame-per-clock-cycle (OFPCC). The frame sampling rate of the beamformer is equal to the clock frequency Fclock≤1/ΔTcpd Hz.An approximately frequency-independent beam shape over an UWB frequency range is obtained by employing a fan filter bank configuration where each subband of the fan filter bank consists of a temporally bandpass frequency-planar beam filter [12]. For the best real-time throughput, each subband of the fan filter bank [12] may be realized using dedicated massively parallel systolic-array processors. However, this technique results in high circuit complexity due to the high degree of parallelism. In this work, we trade throughput for lower circuit complexity by employing a folded architecture for the realization of the fan filter bank. The hardware design approach of folding leads to the time interleaving of multiple 2D filters. Time multiplexing of filter coefficients in folded hardware allows different filters [13] to share arithmetic hardware thereby reducing the number of multipliers and adders. K-times folding results in K-fold utilization of hardware at the cost of a K-fold loss in ULA linear frame-rate [14].


## 2. Review: Beam Filter Design

### 2.1. Review of UWB Beamforming

Beamforming enables array factors (AFs) [15] with steerable directional sensitivities which are used to enhance/reject plane-wave signals [16–22]. Real-time UWB beamforming benefits both engineering and scientific applications such as radar [23–26], navigation [27], wireless communications [18, 28–31], cognitive radio [32, 33], frequency-agile antennas [34], radio astronomy and RF space imaging [35–45], microwave imaging and remote sensing applications [46–48]. Radio telescopes [49–52] such as the square kilometer array (SKA) [53–55] requires algorithms for real-time UWB beamforming (70 MHz–1.4 GHz) with multiple beams [56].Aperture arrays [16–22, 42, 57–65] enable directional enhancement of plane-wave signals [15] (such as signals from deep space, wireless base stations, etc.). The electronically steerable nature of the beam direction and size makes beamforming an essential signal processing task in many applications including radio astronomy [35–45, 49–52, 66–88], radar and navigation [23–27], and broadband wireless communication systems [18, 29–31]. Algorithms for beamforming require low-noise amplification [69–71] of the analog signals from the aperture arrays.In digital beamforming, an array of time synchronous A/D converters [72] leads to the discrete-time signals. Such algorithms are based on either time domain delay-and-sum (DAS) or frequency domain phased array feed (PAF) techniques [16, 19, 73]. In delay-and-sum-based systems, element signals are delayed and added coherently to form a beam. True time delays for each antenna are found for a particular beam direction and inter antenna spacing and are realized as tapped delay lines [74, 75]. In finite impulse response (FIR) beamforming [60, 61, 76, 77], the time delays are implemented as FIR digital filters. This method has high computational intensity compared to the proposed IIR technique. In frequency domain PAF beamforming [57, 58, 65], the digitized DAA signals are converted to frequency domain by evaluating the temporal fast Fourier transform (FFT) [78, 79]. The FFT bins are subsequently multiplied by a complex weight and coherently summed to achieve frequency domain beamforming.


### 2.2. Principle of 2D IIR Beamforming

Figure1 shows the overview of the proposed UWB digital beamforming system. The digital input signals of the proposed massively parallel systolic-array architectures are obtained by amplifying and low-pass filtering the continuous-time (i.e., analog) signals from UWB low-noise antennas. Typical choices for the RF sensing antenna are Vivaldi or BAVA antennas; however, other types of broadband antennas such as biconical antennas may also be employed.Figure 1
RF front-end for a single array element showing antenna, LNA, low-pass filter, and A/D converters. The smart antenna array consists of several such mixed-signal RF circuits for electromagnetic signal sampling at real-time RF rates.UWB beamforming is implemented here using an array of antennas placed at uniform distance alongx-axis. The principle of UWB beam filtering [2, 80] is to enhance the spectrum Wpass(jω1,jω2,ψ) of spatio-temporal plane wave wpass(Δxn1,cΔTsn2,ψ) propagating with a desired direction of arrival (DOA) of ψ while simultaneously attenuating all undesired signals wstop(Δxn1,cΔTsn2,ψ) with spectrum Wstop(jω1,jω2,ψ) which lie outside the passband of the filters.From Nyquist sampling theorem,Δx=cΔTs, where ΔTs is the temporal sampling period and c≈3·108 ms−1 is the speed of light in air [81]. The plane-wave signal of interest is denoted by wpass(·) while interference signals are denoted by wstop(·), respectively. The 2D sampled input signal is of the form
(1)w(n1,n2)=wpass(lk)+∑k=1M‍wstop(lk)+nv(n1,n2),
where lk=-n1Δxsinθk+n2cΔTs,0≤n2, and 0≤n1<N1 corresponding to the number of elements in the ULA. Additive white Gaussian noise (AWGN) in the 2D signal is represented by nv(n1,n2). The spatial DOA [82] is measured from the broadside direction of the ULA and is denoted by θ where 0≤θ≤90° and ψ is the corresponding space-time DOA, 0≤ψ≤45°,
(2)ψk=arctan(ΔxsinθkcΔTs).


### 2.3. Review of 2D Plane-Wave Beam Filters

The 2D IIR plane-wave beam filters can be synthesized [83] using 2D LC network prototypes. The Laplace transfer function derived from the network is found and subsequently converted to the 2D z-domain using the complex map of bilinear transformation. The resulting z-domain transfer function leads to a computable 2D difference equation of the filter enabling real-time digital VLSI realization using RF rate systolic-array processors. Examples of a 1st-order beam filter were previously investigated in [10]. An example of 1st-order 3D IIR cone filter-bank was first proposed in [12].We here propose a 2nd-order plane-wave filter hardware having useful applications as a building block for achieving fan filter banks. A sharper transition required for aperture arrays can be obtained with higher order filters. Further, we propose 3rd-order plane-wave filter hardware and estimate both complexity, quantization noise level, and performance. The sharper transition (roll-off) for 3rd-order filter in comparison to the 2nd-order filter is demonstrated.Let the 1D input-output Laplace transfer function of a classical resistively-terminatedNth-order LC ladder low-pass network shown in Figure 2 be given by
(3)T(s)=Y(s)W(s).
The above equation can be converted to a 2D Laplace equation by applying frequency-planar transformation, s=s1cosψ+s2sinψ to (3) to obtain T(s1,s2):
(4)T(s1,s2)=Y(s1,s2)W(s1,s2).T(s1,s2) is mapped to the 2D z-domain by applying bilinear transformation to get
(5)H(z1,z2)=N(z1,z2)D(z1,z2)=T(s1,s2)|sk=(1-zk-1)/(1+zk-1),k=1,2.
Note that the above design equations are limited to filter passbands that exist in the second and fourth quadrants of the 2D frequency space. Given that filter stability requires all components to be nonnegative [2], for beamforming in quadrants one and three (i.e., -π/2≤θ≤0), we use nonnegative values in each branch impedance and shunt admittance of the prototype while mirroring the input array signal spatially because w(-x,ct)⟺W(-jω1,jω2) such that the passband spectra now fall within quadrants two and four of the frequency space [2].Figure 2
2D passive LRC circuit forNth-order filter.To obtain 2D difference equations, we apply the inverseZ-transform to the above function under ZICs, leading to Nth-order plane-wave beam filter realizations having general form [2]
(6)y(n1,n2)=∑i=0N∑j=0Naijw(n1-i,n2-j)-∑i=0N∑j=0Nbijy(n1-i,n2-j).
For feed-back paths i+j≠0, 0≤n1<N1 and 0≤n2. The frequency response of the Laplace domain prototype is obtained by evaluating T(jω1,jω2). The frequency response of the digital realization is obtained in closed form by evaluating H(ejω1,ejω2) [2]. That is, on the 2D unit bicircle |z1|=|z2|=1. The frequency response may be verified by computing the 2D discrete Fourier transform (DFT) of the unit impulse response h(n1,n2) and comparing with the closed-form response H(ejω1,ejω2).


## 2.1. Review of UWB Beamforming

Beamforming enables array factors (AFs) [15] with steerable directional sensitivities which are used to enhance/reject plane-wave signals [16–22]. Real-time UWB beamforming benefits both engineering and scientific applications such as radar [23–26], navigation [27], wireless communications [18, 28–31], cognitive radio [32, 33], frequency-agile antennas [34], radio astronomy and RF space imaging [35–45], microwave imaging and remote sensing applications [46–48]. Radio telescopes [49–52] such as the square kilometer array (SKA) [53–55] requires algorithms for real-time UWB beamforming (70 MHz–1.4 GHz) with multiple beams [56].Aperture arrays [16–22, 42, 57–65] enable directional enhancement of plane-wave signals [15] (such as signals from deep space, wireless base stations, etc.). The electronically steerable nature of the beam direction and size makes beamforming an essential signal processing task in many applications including radio astronomy [35–45, 49–52, 66–88], radar and navigation [23–27], and broadband wireless communication systems [18, 29–31]. Algorithms for beamforming require low-noise amplification [69–71] of the analog signals from the aperture arrays.In digital beamforming, an array of time synchronous A/D converters [72] leads to the discrete-time signals. Such algorithms are based on either time domain delay-and-sum (DAS) or frequency domain phased array feed (PAF) techniques [16, 19, 73]. In delay-and-sum-based systems, element signals are delayed and added coherently to form a beam. True time delays for each antenna are found for a particular beam direction and inter antenna spacing and are realized as tapped delay lines [74, 75]. In finite impulse response (FIR) beamforming [60, 61, 76, 77], the time delays are implemented as FIR digital filters. This method has high computational intensity compared to the proposed IIR technique. In frequency domain PAF beamforming [57, 58, 65], the digitized DAA signals are converted to frequency domain by evaluating the temporal fast Fourier transform (FFT) [78, 79]. The FFT bins are subsequently multiplied by a complex weight and coherently summed to achieve frequency domain beamforming.


## 2.2. Principle of 2D IIR Beamforming

Figure1 shows the overview of the proposed UWB digital beamforming system. The digital input signals of the proposed massively parallel systolic-array architectures are obtained by amplifying and low-pass filtering the continuous-time (i.e., analog) signals from UWB low-noise antennas. Typical choices for the RF sensing antenna are Vivaldi or BAVA antennas; however, other types of broadband antennas such as biconical antennas may also be employed.Figure 1
RF front-end for a single array element showing antenna, LNA, low-pass filter, and A/D converters. The smart antenna array consists of several such mixed-signal RF circuits for electromagnetic signal sampling at real-time RF rates.UWB beamforming is implemented here using an array of antennas placed at uniform distance alongx-axis. The principle of UWB beam filtering [2, 80] is to enhance the spectrum Wpass(jω1,jω2,ψ) of spatio-temporal plane wave wpass(Δxn1,cΔTsn2,ψ) propagating with a desired direction of arrival (DOA) of ψ while simultaneously attenuating all undesired signals wstop(Δxn1,cΔTsn2,ψ) with spectrum Wstop(jω1,jω2,ψ) which lie outside the passband of the filters.From Nyquist sampling theorem,Δx=cΔTs, where ΔTs is the temporal sampling period and c≈3·108 ms−1 is the speed of light in air [81]. The plane-wave signal of interest is denoted by wpass(·) while interference signals are denoted by wstop(·), respectively. The 2D sampled input signal is of the form
(1)w(n1,n2)=wpass(lk)+∑k=1M‍wstop(lk)+nv(n1,n2),
where lk=-n1Δxsinθk+n2cΔTs,0≤n2, and 0≤n1<N1 corresponding to the number of elements in the ULA. Additive white Gaussian noise (AWGN) in the 2D signal is represented by nv(n1,n2). The spatial DOA [82] is measured from the broadside direction of the ULA and is denoted by θ where 0≤θ≤90° and ψ is the corresponding space-time DOA, 0≤ψ≤45°,
(2)ψk=arctan(ΔxsinθkcΔTs).


## 2.3. Review of 2D Plane-Wave Beam Filters

The 2D IIR plane-wave beam filters can be synthesized [83] using 2D LC network prototypes. The Laplace transfer function derived from the network is found and subsequently converted to the 2D z-domain using the complex map of bilinear transformation. The resulting z-domain transfer function leads to a computable 2D difference equation of the filter enabling real-time digital VLSI realization using RF rate systolic-array processors. Examples of a 1st-order beam filter were previously investigated in [10]. An example of 1st-order 3D IIR cone filter-bank was first proposed in [12].We here propose a 2nd-order plane-wave filter hardware having useful applications as a building block for achieving fan filter banks. A sharper transition required for aperture arrays can be obtained with higher order filters. Further, we propose 3rd-order plane-wave filter hardware and estimate both complexity, quantization noise level, and performance. The sharper transition (roll-off) for 3rd-order filter in comparison to the 2nd-order filter is demonstrated.Let the 1D input-output Laplace transfer function of a classical resistively-terminatedNth-order LC ladder low-pass network shown in Figure 2 be given by
(3)T(s)=Y(s)W(s).
The above equation can be converted to a 2D Laplace equation by applying frequency-planar transformation, s=s1cosψ+s2sinψ to (3) to obtain T(s1,s2):
(4)T(s1,s2)=Y(s1,s2)W(s1,s2).T(s1,s2) is mapped to the 2D z-domain by applying bilinear transformation to get
(5)H(z1,z2)=N(z1,z2)D(z1,z2)=T(s1,s2)|sk=(1-zk-1)/(1+zk-1),k=1,2.
Note that the above design equations are limited to filter passbands that exist in the second and fourth quadrants of the 2D frequency space. Given that filter stability requires all components to be nonnegative [2], for beamforming in quadrants one and three (i.e., -π/2≤θ≤0), we use nonnegative values in each branch impedance and shunt admittance of the prototype while mirroring the input array signal spatially because w(-x,ct)⟺W(-jω1,jω2) such that the passband spectra now fall within quadrants two and four of the frequency space [2].Figure 2
2D passive LRC circuit forNth-order filter.To obtain 2D difference equations, we apply the inverseZ-transform to the above function under ZICs, leading to Nth-order plane-wave beam filter realizations having general form [2]
(6)y(n1,n2)=∑i=0N∑j=0Naijw(n1-i,n2-j)-∑i=0N∑j=0Nbijy(n1-i,n2-j).
For feed-back paths i+j≠0, 0≤n1<N1 and 0≤n2. The frequency response of the Laplace domain prototype is obtained by evaluating T(jω1,jω2). The frequency response of the digital realization is obtained in closed form by evaluating H(ejω1,ejω2) [2]. That is, on the 2D unit bicircle |z1|=|z2|=1. The frequency response may be verified by computing the 2D discrete Fourier transform (DFT) of the unit impulse response h(n1,n2) and comparing with the closed-form response H(ejω1,ejω2).


## 3. Systolic-Array Architecture of 2D IIR Filters

### 3.1. Difference Equations

In our example design for 2nd-order plane-wave filter shown in Figure3, we employ parameters Rs=1, L1=1.4142, C2=1.4142 [83]. For our example of 3rd-order beam filter shown in Figure 4, the design parameters are Rs=1, L1=1, C2=2, L3=1 and the desired DOA is ψ=10° [83]. Table 1 gives the feedback coefficients of the filter for a 1st-order, 2nd-order, and 3rd-order plane-wave beam filters [84].Table 1
Filter design equations showing algebraically defined filter coefficients as functions of LC ladder prototype network parameters and steered angletanψ=sinθ. Here, · refers to dot product between vectors.
Orderof filter
Arbitrary coefficients
Coefficients
e
m
b
i
j
1
b
i
j
=
e
^
·
β
^
i
j
,
b
00
=
0
L
i
=
L
cos
ψ,
e
^
=
(
e
1
,
e
2
,
e
3
)
/
ϵ
L
j
=
L
sin
ψ,
ϵ
=
e
1
+
e
2
+
e
3
e
1
=
1
+
R
s,
β
01
=
(
1,1
,
-
1
)
e
2
=
L
i
,
e
3
=
L
j
β
10
=
(
1
,
-
1,1
)
β
11
=
(
1
,
-
1
,
-
1
)
2
b
i
j
=
e
^
·
β
^
i
j
,
b
00
=
0
L
i
=
L
1
cos
ψ,
e
^
=
(
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
)
/
ϵ
L
j
=
L
1
sin
ψ,
ϵ
=
e
1
+
e
2
+
e
3
+
e
4
+
e
4
+
e
6
C
i
=
C
2
cos
ψ,
β
01
=
(
2,2
,
0,2
,
-
2,0
)
C
j
=
C
2
sin
ψ.
β
02
=
(
1,1
,
-
1,1
,
1
,
-
1
)
e
1
=
1
+
R
s,
β
10
=
(
2,0
,
2
,
-
2,2
,
0
)
e
2
=
L
i
+
R
s
C
i,
β
11
=
(
4,0
,
0
,
-
4
,
-
4,0
)
e
3
=
L
j
+
R
s
C
j,
β
12
=
(
2,0
,
-
2
,
-
2,2
,
0
)
e
4
=
L
i
C
i
,
e
5
=
L
j
C
j,
β
20
=
(
1
,
-
1,1
,
1,1
,
-
1
)
e
6
=
L
i
C
j
+
L
j
C
i.
β
21
=
(
2
,
-
2,0
,
2
,
-
2,0
)
β
22
=
(
1
,
-
1
,
-
1,1
,
1,1
)
3
b
i
j
=
e
^
·
β
^
i
j
,
b
00
=
0,
e
^
=
(
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
,
e
7
,
e
8
,
e
9
,
e
10
)
/
ϵ,
L
i
=
L
1
cos
ψ,
ϵ
=
e
1
+
e
2
+
e
3
+
e
4
+
e
5
+
e
6
+
e
7
+
e
8
+
e
9
+
e
10
L
j
=
L
1
sin
ψ,
β
01
=
(
3,3
,
1,3
,
-
1,1
,
3
,
-
3,1
,
-
1
)
C
i
=
C
2
cos
ψ,
β
02
=
(
3,3
,
-
1,3
,
-
1
,
-
1,3
,
3
,
-
1
,
-
1
)
C
j
=
C
2
sin
ψ,
β
03
=
(
1,1
,
-
1,1
,
1
,
-
1,1
,
-
1
,
-
1,1
)
L
k
=
L
3
cos
ψ,
β
10
=
(
3,1
,
3
,
-
1,3
,
1
,
-
3,3
,
-
1,1
)
L
l
=
L
3
sin
ψ,
β
11
=
(
9,3
,
3
,
-
3
,
-
3,1
,
-
9
,
-
9
,
-
1
,
-
1
)
e
1
=
1
+
R
s,
β
12
=
(
9,3
,
-
3
,
-
3
,
-
3
,
-
1
,
-
9,9
,
1
,
-
1
)
e
2
=
L
i
+
L
k
+
R
s
C
i,
β
13
=
(
3,1
,
-
3
,
-
1,3
,
-
1
,
-
3
,
-
3,1
,
1
)
e
3
=
L
j
+
L
l
+
R
s
C
j,
β
20
=
(
3
,
-
1,3
,
-
1,3
,
-
1,3
,
3
,
-
1
,
-
1
)
e
4
=
L
i
C
i
+
L
k
C
i,
β
21
=
(
9
,
-
3,3
,
-
3
,
-
3
,
-
1,9
,
-
9
,
-
1,1
)
e
5
=
L
j
C
j
+
L
l
C
j,
β
22
=
(
9
,
-
3
,
-
3
,
-
3
,
-
3,1
,
9,9
,
1,1
)
e
6
=
L
i
C
j
+
L
j
C
i
+
L
k
C
j
+
L
l
C
i,
β
23
=
(
3
,
-
1
,
-
3
,
-
1,3
,
1,3
,
-
3,1
,
-
1
)
e
7
=
L
i
L
k
C
i
,
e
8
=
L
j
L
l
C
j,
β
30
=
(
1
,
-
1,1
,
1,1
,
-
1
,
-
1,1
,
1
,
-
1
)
e
9
=
L
i
L
k
C
j
+
L
i
L
l
C
i
+
L
j
L
k
C
i,
β
31
=
(
3
,
-
3,1
,
3
,
-
1
,
-
1
,
-
3
,
-
3,1
,
1
)
e
10
=
L
i
C
i
L
l
+
L
j
L
k
C
j
+
C
i
L
j
L
l.
β
32
=
(
3
,
-
3
,
-
1,3
,
-
1,1
,
-
3,3
,
-
1,1
)
β
33
=
(
1
,
-
1
,
-
1,1
,
1,1
,
-
1
,
-
1
,
-
1
,
-
1
)Figure 3
2D passive LRC circuit for 2nd-order filter.Figure 4
2D passive LRC circuit for 3rd-order filter.


### 3.2. Partially Separable Signal Flow Graphs

In order to reduce digital implementation complexity, we first separate the 2Dz-domain transfer function of the beam filters into separable and nonseparable subfilters. These subfilters correspond to spatial, temporal, and spatio-temporal prototype networks. Therefore, the transfer functions are cascaded to form the final 2D filter as shown in Figure 5. Following [10] to higher-order filters, we propose that the nonseparable function H1(z1,z2) be realized employing a systolic-array consisting of N1 similar parallel processing core modules (PPCMs) which are interconnected to each other such that spatio-temporal feed-forward and feedback paths required for recursive computation of the filter are realized using 2D difference equations.Figure 5
Cascaded form of 2DNth-order beam filter.The input RF waves received by the each antenna in the ULA is passed through LNAs, low pass filtered, samples, and quantized within each ADC. The sampled digital signals are connected to the PPCM input ports. The corresponding outputs are sent throughH2(z1) and H3(z2) as shown in Figure 1. The function H2(z1) is implemented as filter with spatial delays and H3(z2) is implemented using temporal delays:
(7)H(z1,z2)=H1(z1,z2)H2(z1)H3(z2),H1(z1,z2)=11+∑i=0N∑j=0Nbijz1-iz2-j,H2(z1)=(1+z1-1)N,H3(z2)=(1+z2-1)N.
Zero-initial conditions (ZICs) of the filter along both discrete space and time dimensions are defined by (8a)w(-Δx,cΔTsn2)=w(Δxn1,-cΔTs)≡0,(8b)y(-Δx,cΔTsn2)=y(Δxn1,-cΔTs)≡0.The temporal ZICs [10, 11] are provided to the design by preloading the input values with zeros. Spatial ZICs are provided by connecting constant value zero as previous state inputs to the first PPCM. The function H1(z1,z2) for an Nth-order plane-wave beam filter given in (7) is implemented as N1 similar interconnected PPCMs. Realization of H1(z1,z2) for a 2nd-order filter is shown in Figure 6 and for 3rd-order filter as shown in Figure 7.Figure 6
Architecture ofH1,m′(n1,z2) of 2D 2nd-order beam filter with 3-stage scattered look-ahead pipelining.Figure 7
Architecture ofH1,m′(n1,z2) of 2D 3rd-order beam filter with 3-stage scattered look-ahead pipelining.


### 3.3. Fixed-Point Arithmetic and Quantization Effects

Signed fractional numbers are quantized in a finite precision digital representation, which in this case is based on the two’s complement format. Hence, we quantize the coefficients of 2D digital IIR filters using signed fixed-point arithmetic. The size of registers is designed such that(W,D) being the size of the input signal where W is the total size of the two’s complement number and D is the binary point location counted from the rightmost position. Also, we use (Wc,Dc) as the size of the coefficients of the filter. The fixed-point finite register size of multipliers has been designed at (W+Wc,D+Dc). The size of each digital signal representation after each multiplier and adder has been marked at every point in the realization of H1,m′(n1,z2). The output signal of a given PPCM is provided as one of the inputs to the next PPCM; this requires that the size of the output signal of each PPCM be the same as the size of the input port in the neighbouring PPCM.


### 3.4. Beam Sensitivity to Filter Coefficient Precision

First-order sensitivity gives a measure of error associated with perturbations in coefficients of the filter. The sensitivity of the 2D beam transfer function of the filter due to changes in the values of the coefficients that results due to quantization is studied next. We consider 1st-order sensitivity function to find the magnitude error of the design [85]. The sensitivity function for 2D beam function H(z1,z2) is given by
(9)SbijH(ejω1,ejω2)=-bijD(z1,z2)·z1-iz2-j.
Sensitivity helps to find relative error in transfer function with respect to a coefficient. If bij is a coefficient of the filter, then the relative error in the transfer function of the filter due to perturbations in bij is given by
(10)ΔHH=SbijH(Δbijbij).
The gain sensitivity in |H(ejω1,ejω2)| is computed using
(11)SbijG(ω1,ω2)=Real(Sbij|H(ejω1,ejω2)|).
The gain sensitivity in |H(ejω1,ejω2)| due to fixed-point errors in all the coefficients of the filter is given by the summation of the gain sensitivities with respect to each coefficient [85]. We consider nonuniform error in the coefficients given by ϵi. The relative error in H(z1,z2) with respect to different sizes of coefficients for 2nd-order and 3rd-order filters is shown in Figure 8. Relative error [85] in |H(z1,z2)| is calculated as follows:
(12)ΔHH=∑i=0N-1‍∑j=0N-1‍Real(Sbij|H(ejω1,ejω2)|)Δbijbij,
where i+j≠0. From Figure 8 for the usable frequency range ω2≤π/2, we observe that, for 2nd-order beamformers, Wc=15-bit precision in the filter coefficients leads to beam accuracy within 3% which improves to better than 0.1% for Wc=20-bit precision. For 3rd-order beamformers, Wc=19-bit precision in the filter coefficients leads to 3% accuracy, which improves to better than 0.1% accuracy when the filter coefficient precision is increased to Wc=23-bits.Relative error inH(z1,z2) with respect to the respective coefficients of 2nd-order and 3rd-order plane-wave filters.
(a)
Relative error inH of 2nd-order filter for Wc=20
(b)
Relative error inH of 2nd-order filter for Wc=15
(c)
Relative error inH of 3rd-order filter for Wc=23
(d)
Relative error inH of 3rd-order filter for Wc=19


### 3.5. Internal Register/Word Sizes

The word length of the signal at the feedback loop needs to be truncated. To maintain a minimum error, the size of the input signal has been decided based on the maximum value of the output signal. The design has been simulated for various word sizes and compared to the 64-bit precision Matlab outputs. For this experiment, we provided an impulsive UWB signal as an input to the filter because the operation and accuracy of the filter can be tested from the impulse response of the filter.Tables2 and 3 show the variation of quantization noise with the word size of the coefficients of the filter and the input size. In general, increased precision in the recursive spatio-temporal feedback sections leads to larger VLSI area and higher power consumption with lower speed due to larger CPDs. A compromise must be found depending on the needs of the target application that balances power, speed, accuracy, and chip area.Table 2
Calculation of error energy for unit impulse input for different output sizes of 2nd-order beam filter.
(
W
,
D
)
(
W
c
,
D
c
)
(20, −17)
(19, −16)
(18, −15)
(17, −14)
(16, −13)
(15, −12)
(27, −17)
0.0104
0.0425
0.2292
0.1107
4.0944
5.8795
(26, −16)
0.0321
0.0831
0.2759
0.1431
4.7827
5.9277
(25, −15)
0.1173
0.2269
0.3645
0.2335
5.1569
5.8628
(24, −14)
0.4520
0.5424
0.6842
0.7018
5.5859
6.1265
(23, −13)
1.6935
2.1197
1.9373
1.8941
6.3689
7.8321
(22, −12)
7.3510
6.8815
7.1973
7.7967
12.5538
12.7566
(21,−11)
25.4179
31.9657
29.9076
35.5036
37.7186
40.0737Table 3
Calculation of unnormalizedL2-error energy for unit impulse input for different output sizes of 3rd-order beam filter.
(W,D)
(
W
c
,
D
c
)
(
23
,
-
19
)
(
22
,
-
18
)
(
21
,
-
17
)
(
20
,
-
16
)
(
19
,
-
15
)
(
34
,
-
19
)
0.0496
0.1097
0.3805
12.7429
3.4765
(
33
,
-
18
)
0.0506
0.1148
0.3799
12.9468
3.4434
(
32
,
-
17
)
0.0544
0.1180
0.3922
12.4035
3.5761
(
31
,
-
16
)
0.0828
0.1381
0.4739
13.5182
3.4513
(
30
,
-
15
)
0.1786
0.2397
0.7910
14.0064
3.8871
(
29
,
-
14
)
0.5750
0.7209
1.6789
16.1490
4.6594


### 3.6. Look-Ahead Speed Optimization

The speed of the designs is maximized using fine-grain pipelining. Multilevel pipelining, that is, both inter-PPCM pipelining and intrapipelining, is used for maximum performance. Intrapipelining refers to internal optimization within the PPCMs. The feedback loop of an IIR filter inside the PPCM can be pipelined using look-ahead pipelining. Inter-PPCM pipelining refers to the pipelining of signal paths outside the PPCMs. The application of look-ahead pipelining to non-separable 2D IIR digital filters was first proposed in [10]. Here, we pipeline each feedback loop of the proposed filters using stable scattered look-ahead pipelining (SLA) [11, 86, 87]. In SLA, additional zeros and poles which are at same angular distance as the original poles are introduced into the transfer function, enabling the CPD of the filter to be reduced.If the denominator of the transfer function is in the form ofD(z),
(13)D(z)=∏k=1N‍(1-piz-1).
The general equation of M-stage SLA pipelining [86] is given by
(14)H(z)=N(z)D(z)=N(z)∏i=1N∏k=1M-1(1-piej2πk/Mz-1)∏i=1N∏k=0M-1(1-piej2πk/Mz-1)=N′(z)D′(zM).
The feedback loop for 2nd-order beam filter as shown in the following equation is pipelined for 3 stages in our design example:
(15)H1,m(n1,z2)=11+b01z2-1+b02z2-2.
Transfer function for 3-stage look-ahead pipelining of feedback loop is given by
(16)H1,m′(n1,z2)=1+A11z2-1+A12z2-2+A13z2-3+A14z2-41+B11z2-3+B12z2-6,
where
(17)A11=-b01,A12=(b012-b02),A13=-b01b02,A14=b022,B11=-(-b013+3b01b02),B12=b023.
For the 3rd order, the feedback loop is described using the following equation:
(18)H1,m(n1,z2)=11+b01z2-1+b02z2-2+b03z2-3.
The transfer function for 3-stage look-ahead pipelining of feedback loop is given by
(19)H1,m′(n1,z2)=(1+A11z2-1+A12z2-2+A13z2-3)(1+B11z2-3+B12z2-6+B13z2-9)+(A14z2-4+A15z2-5+A16z2-6)(1+B11z2-3+B12z2-6+B13z2-9),
where
(20)A11=-b01,A12=(b012-b02),A13=2b03-b01b02,A14=b022-b02b03,A15=-b02b03,A16=b032,B11=b013-3b01b02+3b03,B12=b023+2b01b03+3b032,B13=b033.Figure9 shows the realization of H2(z1) of 2nd-order beam filter. H2(z1) is a spatial function. Hence, it is realized as sum of three consecutive 1D outputs given by the PPCMs of H1(z1,z2). H3(z2) is temporal function and it is realized as consecutive 1D filters as shown in Figure 10. Similar are the realizations for the 3rd-order filter, as shown in Figure 11 and described by Figure 12.Figure 9
Architecture ofH2(z1) of 2D 2nd-order beam filter.Figure 10
Architecture ofH3(z2) of 2D 2nd-order beam filter.Figure 11
Architecture ofH2(z1) of 2D 3rd-order beam filter.Figure 12
Architecture ofH3(z2) of 2D 3rd-order beam filter.


### 3.7. Time Multiplexing for Folded Architectures

Time-multiplexed systolic-array designs are useful for the design of fan filters [88] whose transfer function is given in the following. Figure 13 shows how fan filters are designed using time-multiplexed filters and are described using [12]
(21)HFan(z1,z2)=∑k=1K‍Hbeam,k(z1,z2)TFIR,k(z2),
where TFIR,k(z2) are subbands of a perfect reconstruction FIR bandpass filter bank.Figure 13
Overview of aK-time multiplexed filter bank architecture [12].Time multiplexing ofK-filters needs the inputs to be upsampled by K∈Z+ with copying [88, 89]. The input signal consists of K samples pertaining to each original sample and are passed through the time-multiplexed filter. In Figure 14, we provide an overview of signal flow in time-multiplexed design of the K beam filters. These filters give K-outputs which are to be demultiplexed before being applied to the fan filter bank perfect reconstruction FIR bandpass filters [12]. The time multiplexing of the folded architecture caused each unit delay to be increased to
(22)ΔT=KΔTclk,
where K is the number of inputs (K=4, in the example provided) and ΔTclk is the clock delay.Figure 14
Overview of time multiplexingK-beam filters.The signal flow inside the time-multiplexed PPCM is such that the coefficients of feedback terms ofK filters are given to a two-input multiplier through a commutating multiplexer. Input signal w(Δxn1,cΔTsn2) is given to the other input of multiplier. A counter is used to select feedback coefficient given to multiplexer. We have designed the filter bank with four filters. Hence, the counter runs from 0 to 3 and restarts. Therefore, when there exists K filter coefficients, the required counter must continuously count up 0 to log2K. The critical-path delay reduces as the architecture is fine-grain pipelined with SLA pipelining for the feedback path. Figure 15 shows each fully multiplexed multiplier circuit (FMMC) design for beam filters. 2nd-order and 3rd-order time-multiplexed designs are as shown in Figures 16 and 17, respectively, with ΔT as in (22).Figure 15
Fully multiplexed multiplier circuit forK-time-multiplexed filters.Figure 16
Time-multiplexed PPCM of 2nd-order beam filter bank with 3-stage scattered look-ahead pipelining forK-time-multiplexed beam filters.Figure 17
Time-multiplexed PPCM of 3rd-order beam filter bank with 3-stage scattered look-ahead pipelining forK-time-multiplexed beam filters.


## 3.1. Difference Equations

In our example design for 2nd-order plane-wave filter shown in Figure3, we employ parameters Rs=1, L1=1.4142, C2=1.4142 [83]. For our example of 3rd-order beam filter shown in Figure 4, the design parameters are Rs=1, L1=1, C2=2, L3=1 and the desired DOA is ψ=10° [83]. Table 1 gives the feedback coefficients of the filter for a 1st-order, 2nd-order, and 3rd-order plane-wave beam filters [84].Table 1
Filter design equations showing algebraically defined filter coefficients as functions of LC ladder prototype network parameters and steered angletanψ=sinθ. Here, · refers to dot product between vectors.
Orderof filter
Arbitrary coefficients
Coefficients
e
m
b
i
j
1
b
i
j
=
e
^
·
β
^
i
j
,
b
00
=
0
L
i
=
L
cos
ψ,
e
^
=
(
e
1
,
e
2
,
e
3
)
/
ϵ
L
j
=
L
sin
ψ,
ϵ
=
e
1
+
e
2
+
e
3
e
1
=
1
+
R
s,
β
01
=
(
1,1
,
-
1
)
e
2
=
L
i
,
e
3
=
L
j
β
10
=
(
1
,
-
1,1
)
β
11
=
(
1
,
-
1
,
-
1
)
2
b
i
j
=
e
^
·
β
^
i
j
,
b
00
=
0
L
i
=
L
1
cos
ψ,
e
^
=
(
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
)
/
ϵ
L
j
=
L
1
sin
ψ,
ϵ
=
e
1
+
e
2
+
e
3
+
e
4
+
e
4
+
e
6
C
i
=
C
2
cos
ψ,
β
01
=
(
2,2
,
0,2
,
-
2,0
)
C
j
=
C
2
sin
ψ.
β
02
=
(
1,1
,
-
1,1
,
1
,
-
1
)
e
1
=
1
+
R
s,
β
10
=
(
2,0
,
2
,
-
2,2
,
0
)
e
2
=
L
i
+
R
s
C
i,
β
11
=
(
4,0
,
0
,
-
4
,
-
4,0
)
e
3
=
L
j
+
R
s
C
j,
β
12
=
(
2,0
,
-
2
,
-
2,2
,
0
)
e
4
=
L
i
C
i
,
e
5
=
L
j
C
j,
β
20
=
(
1
,
-
1,1
,
1,1
,
-
1
)
e
6
=
L
i
C
j
+
L
j
C
i.
β
21
=
(
2
,
-
2,0
,
2
,
-
2,0
)
β
22
=
(
1
,
-
1
,
-
1,1
,
1,1
)
3
b
i
j
=
e
^
·
β
^
i
j
,
b
00
=
0,
e
^
=
(
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
,
e
7
,
e
8
,
e
9
,
e
10
)
/
ϵ,
L
i
=
L
1
cos
ψ,
ϵ
=
e
1
+
e
2
+
e
3
+
e
4
+
e
5
+
e
6
+
e
7
+
e
8
+
e
9
+
e
10
L
j
=
L
1
sin
ψ,
β
01
=
(
3,3
,
1,3
,
-
1,1
,
3
,
-
3,1
,
-
1
)
C
i
=
C
2
cos
ψ,
β
02
=
(
3,3
,
-
1,3
,
-
1
,
-
1,3
,
3
,
-
1
,
-
1
)
C
j
=
C
2
sin
ψ,
β
03
=
(
1,1
,
-
1,1
,
1
,
-
1,1
,
-
1
,
-
1,1
)
L
k
=
L
3
cos
ψ,
β
10
=
(
3,1
,
3
,
-
1,3
,
1
,
-
3,3
,
-
1,1
)
L
l
=
L
3
sin
ψ,
β
11
=
(
9,3
,
3
,
-
3
,
-
3,1
,
-
9
,
-
9
,
-
1
,
-
1
)
e
1
=
1
+
R
s,
β
12
=
(
9,3
,
-
3
,
-
3
,
-
3
,
-
1
,
-
9,9
,
1
,
-
1
)
e
2
=
L
i
+
L
k
+
R
s
C
i,
β
13
=
(
3,1
,
-
3
,
-
1,3
,
-
1
,
-
3
,
-
3,1
,
1
)
e
3
=
L
j
+
L
l
+
R
s
C
j,
β
20
=
(
3
,
-
1,3
,
-
1,3
,
-
1,3
,
3
,
-
1
,
-
1
)
e
4
=
L
i
C
i
+
L
k
C
i,
β
21
=
(
9
,
-
3,3
,
-
3
,
-
3
,
-
1,9
,
-
9
,
-
1,1
)
e
5
=
L
j
C
j
+
L
l
C
j,
β
22
=
(
9
,
-
3
,
-
3
,
-
3
,
-
3,1
,
9,9
,
1,1
)
e
6
=
L
i
C
j
+
L
j
C
i
+
L
k
C
j
+
L
l
C
i,
β
23
=
(
3
,
-
1
,
-
3
,
-
1,3
,
1,3
,
-
3,1
,
-
1
)
e
7
=
L
i
L
k
C
i
,
e
8
=
L
j
L
l
C
j,
β
30
=
(
1
,
-
1,1
,
1,1
,
-
1
,
-
1,1
,
1
,
-
1
)
e
9
=
L
i
L
k
C
j
+
L
i
L
l
C
i
+
L
j
L
k
C
i,
β
31
=
(
3
,
-
3,1
,
3
,
-
1
,
-
1
,
-
3
,
-
3,1
,
1
)
e
10
=
L
i
C
i
L
l
+
L
j
L
k
C
j
+
C
i
L
j
L
l.
β
32
=
(
3
,
-
3
,
-
1,3
,
-
1,1
,
-
3,3
,
-
1,1
)
β
33
=
(
1
,
-
1
,
-
1,1
,
1,1
,
-
1
,
-
1
,
-
1
,
-
1
)Figure 3
2D passive LRC circuit for 2nd-order filter.Figure 4
2D passive LRC circuit for 3rd-order filter.


## 3.2. Partially Separable Signal Flow Graphs

In order to reduce digital implementation complexity, we first separate the 2Dz-domain transfer function of the beam filters into separable and nonseparable subfilters. These subfilters correspond to spatial, temporal, and spatio-temporal prototype networks. Therefore, the transfer functions are cascaded to form the final 2D filter as shown in Figure 5. Following [10] to higher-order filters, we propose that the nonseparable function H1(z1,z2) be realized employing a systolic-array consisting of N1 similar parallel processing core modules (PPCMs) which are interconnected to each other such that spatio-temporal feed-forward and feedback paths required for recursive computation of the filter are realized using 2D difference equations.Figure 5
Cascaded form of 2DNth-order beam filter.The input RF waves received by the each antenna in the ULA is passed through LNAs, low pass filtered, samples, and quantized within each ADC. The sampled digital signals are connected to the PPCM input ports. The corresponding outputs are sent throughH2(z1) and H3(z2) as shown in Figure 1. The function H2(z1) is implemented as filter with spatial delays and H3(z2) is implemented using temporal delays:
(7)H(z1,z2)=H1(z1,z2)H2(z1)H3(z2),H1(z1,z2)=11+∑i=0N∑j=0Nbijz1-iz2-j,H2(z1)=(1+z1-1)N,H3(z2)=(1+z2-1)N.
Zero-initial conditions (ZICs) of the filter along both discrete space and time dimensions are defined by (8a)w(-Δx,cΔTsn2)=w(Δxn1,-cΔTs)≡0,(8b)y(-Δx,cΔTsn2)=y(Δxn1,-cΔTs)≡0.The temporal ZICs [10, 11] are provided to the design by preloading the input values with zeros. Spatial ZICs are provided by connecting constant value zero as previous state inputs to the first PPCM. The function H1(z1,z2) for an Nth-order plane-wave beam filter given in (7) is implemented as N1 similar interconnected PPCMs. Realization of H1(z1,z2) for a 2nd-order filter is shown in Figure 6 and for 3rd-order filter as shown in Figure 7.Figure 6
Architecture ofH1,m′(n1,z2) of 2D 2nd-order beam filter with 3-stage scattered look-ahead pipelining.Figure 7
Architecture ofH1,m′(n1,z2) of 2D 3rd-order beam filter with 3-stage scattered look-ahead pipelining.


## 3.3. Fixed-Point Arithmetic and Quantization Effects

Signed fractional numbers are quantized in a finite precision digital representation, which in this case is based on the two’s complement format. Hence, we quantize the coefficients of 2D digital IIR filters using signed fixed-point arithmetic. The size of registers is designed such that(W,D) being the size of the input signal where W is the total size of the two’s complement number and D is the binary point location counted from the rightmost position. Also, we use (Wc,Dc) as the size of the coefficients of the filter. The fixed-point finite register size of multipliers has been designed at (W+Wc,D+Dc). The size of each digital signal representation after each multiplier and adder has been marked at every point in the realization of H1,m′(n1,z2). The output signal of a given PPCM is provided as one of the inputs to the next PPCM; this requires that the size of the output signal of each PPCM be the same as the size of the input port in the neighbouring PPCM.


## 3.4. Beam Sensitivity to Filter Coefficient Precision

First-order sensitivity gives a measure of error associated with perturbations in coefficients of the filter. The sensitivity of the 2D beam transfer function of the filter due to changes in the values of the coefficients that results due to quantization is studied next. We consider 1st-order sensitivity function to find the magnitude error of the design [85]. The sensitivity function for 2D beam function H(z1,z2) is given by
(9)SbijH(ejω1,ejω2)=-bijD(z1,z2)·z1-iz2-j.
Sensitivity helps to find relative error in transfer function with respect to a coefficient. If bij is a coefficient of the filter, then the relative error in the transfer function of the filter due to perturbations in bij is given by
(10)ΔHH=SbijH(Δbijbij).
The gain sensitivity in |H(ejω1,ejω2)| is computed using
(11)SbijG(ω1,ω2)=Real(Sbij|H(ejω1,ejω2)|).
The gain sensitivity in |H(ejω1,ejω2)| due to fixed-point errors in all the coefficients of the filter is given by the summation of the gain sensitivities with respect to each coefficient [85]. We consider nonuniform error in the coefficients given by ϵi. The relative error in H(z1,z2) with respect to different sizes of coefficients for 2nd-order and 3rd-order filters is shown in Figure 8. Relative error [85] in |H(z1,z2)| is calculated as follows:
(12)ΔHH=∑i=0N-1‍∑j=0N-1‍Real(Sbij|H(ejω1,ejω2)|)Δbijbij,
where i+j≠0. From Figure 8 for the usable frequency range ω2≤π/2, we observe that, for 2nd-order beamformers, Wc=15-bit precision in the filter coefficients leads to beam accuracy within 3% which improves to better than 0.1% for Wc=20-bit precision. For 3rd-order beamformers, Wc=19-bit precision in the filter coefficients leads to 3% accuracy, which improves to better than 0.1% accuracy when the filter coefficient precision is increased to Wc=23-bits.Relative error inH(z1,z2) with respect to the respective coefficients of 2nd-order and 3rd-order plane-wave filters.
(a)
Relative error inH of 2nd-order filter for Wc=20
(b)
Relative error inH of 2nd-order filter for Wc=15
(c)
Relative error inH of 3rd-order filter for Wc=23
(d)
Relative error inH of 3rd-order filter for Wc=19


## 3.5. Internal Register/Word Sizes

The word length of the signal at the feedback loop needs to be truncated. To maintain a minimum error, the size of the input signal has been decided based on the maximum value of the output signal. The design has been simulated for various word sizes and compared to the 64-bit precision Matlab outputs. For this experiment, we provided an impulsive UWB signal as an input to the filter because the operation and accuracy of the filter can be tested from the impulse response of the filter.Tables2 and 3 show the variation of quantization noise with the word size of the coefficients of the filter and the input size. In general, increased precision in the recursive spatio-temporal feedback sections leads to larger VLSI area and higher power consumption with lower speed due to larger CPDs. A compromise must be found depending on the needs of the target application that balances power, speed, accuracy, and chip area.Table 2
Calculation of error energy for unit impulse input for different output sizes of 2nd-order beam filter.
(
W
,
D
)
(
W
c
,
D
c
)
(20, −17)
(19, −16)
(18, −15)
(17, −14)
(16, −13)
(15, −12)
(27, −17)
0.0104
0.0425
0.2292
0.1107
4.0944
5.8795
(26, −16)
0.0321
0.0831
0.2759
0.1431
4.7827
5.9277
(25, −15)
0.1173
0.2269
0.3645
0.2335
5.1569
5.8628
(24, −14)
0.4520
0.5424
0.6842
0.7018
5.5859
6.1265
(23, −13)
1.6935
2.1197
1.9373
1.8941
6.3689
7.8321
(22, −12)
7.3510
6.8815
7.1973
7.7967
12.5538
12.7566
(21,−11)
25.4179
31.9657
29.9076
35.5036
37.7186
40.0737Table 3
Calculation of unnormalizedL2-error energy for unit impulse input for different output sizes of 3rd-order beam filter.
(W,D)
(
W
c
,
D
c
)
(
23
,
-
19
)
(
22
,
-
18
)
(
21
,
-
17
)
(
20
,
-
16
)
(
19
,
-
15
)
(
34
,
-
19
)
0.0496
0.1097
0.3805
12.7429
3.4765
(
33
,
-
18
)
0.0506
0.1148
0.3799
12.9468
3.4434
(
32
,
-
17
)
0.0544
0.1180
0.3922
12.4035
3.5761
(
31
,
-
16
)
0.0828
0.1381
0.4739
13.5182
3.4513
(
30
,
-
15
)
0.1786
0.2397
0.7910
14.0064
3.8871
(
29
,
-
14
)
0.5750
0.7209
1.6789
16.1490
4.6594


## 3.6. Look-Ahead Speed Optimization

The speed of the designs is maximized using fine-grain pipelining. Multilevel pipelining, that is, both inter-PPCM pipelining and intrapipelining, is used for maximum performance. Intrapipelining refers to internal optimization within the PPCMs. The feedback loop of an IIR filter inside the PPCM can be pipelined using look-ahead pipelining. Inter-PPCM pipelining refers to the pipelining of signal paths outside the PPCMs. The application of look-ahead pipelining to non-separable 2D IIR digital filters was first proposed in [10]. Here, we pipeline each feedback loop of the proposed filters using stable scattered look-ahead pipelining (SLA) [11, 86, 87]. In SLA, additional zeros and poles which are at same angular distance as the original poles are introduced into the transfer function, enabling the CPD of the filter to be reduced.If the denominator of the transfer function is in the form ofD(z),
(13)D(z)=∏k=1N‍(1-piz-1).
The general equation of M-stage SLA pipelining [86] is given by
(14)H(z)=N(z)D(z)=N(z)∏i=1N∏k=1M-1(1-piej2πk/Mz-1)∏i=1N∏k=0M-1(1-piej2πk/Mz-1)=N′(z)D′(zM).
The feedback loop for 2nd-order beam filter as shown in the following equation is pipelined for 3 stages in our design example:
(15)H1,m(n1,z2)=11+b01z2-1+b02z2-2.
Transfer function for 3-stage look-ahead pipelining of feedback loop is given by
(16)H1,m′(n1,z2)=1+A11z2-1+A12z2-2+A13z2-3+A14z2-41+B11z2-3+B12z2-6,
where
(17)A11=-b01,A12=(b012-b02),A13=-b01b02,A14=b022,B11=-(-b013+3b01b02),B12=b023.
For the 3rd order, the feedback loop is described using the following equation:
(18)H1,m(n1,z2)=11+b01z2-1+b02z2-2+b03z2-3.
The transfer function for 3-stage look-ahead pipelining of feedback loop is given by
(19)H1,m′(n1,z2)=(1+A11z2-1+A12z2-2+A13z2-3)(1+B11z2-3+B12z2-6+B13z2-9)+(A14z2-4+A15z2-5+A16z2-6)(1+B11z2-3+B12z2-6+B13z2-9),
where
(20)A11=-b01,A12=(b012-b02),A13=2b03-b01b02,A14=b022-b02b03,A15=-b02b03,A16=b032,B11=b013-3b01b02+3b03,B12=b023+2b01b03+3b032,B13=b033.Figure9 shows the realization of H2(z1) of 2nd-order beam filter. H2(z1) is a spatial function. Hence, it is realized as sum of three consecutive 1D outputs given by the PPCMs of H1(z1,z2). H3(z2) is temporal function and it is realized as consecutive 1D filters as shown in Figure 10. Similar are the realizations for the 3rd-order filter, as shown in Figure 11 and described by Figure 12.Figure 9
Architecture ofH2(z1) of 2D 2nd-order beam filter.Figure 10
Architecture ofH3(z2) of 2D 2nd-order beam filter.Figure 11
Architecture ofH2(z1) of 2D 3rd-order beam filter.Figure 12
Architecture ofH3(z2) of 2D 3rd-order beam filter.


## 3.7. Time Multiplexing for Folded Architectures

Time-multiplexed systolic-array designs are useful for the design of fan filters [88] whose transfer function is given in the following. Figure 13 shows how fan filters are designed using time-multiplexed filters and are described using [12]
(21)HFan(z1,z2)=∑k=1K‍Hbeam,k(z1,z2)TFIR,k(z2),
where TFIR,k(z2) are subbands of a perfect reconstruction FIR bandpass filter bank.Figure 13
Overview of aK-time multiplexed filter bank architecture [12].Time multiplexing ofK-filters needs the inputs to be upsampled by K∈Z+ with copying [88, 89]. The input signal consists of K samples pertaining to each original sample and are passed through the time-multiplexed filter. In Figure 14, we provide an overview of signal flow in time-multiplexed design of the K beam filters. These filters give K-outputs which are to be demultiplexed before being applied to the fan filter bank perfect reconstruction FIR bandpass filters [12]. The time multiplexing of the folded architecture caused each unit delay to be increased to
(22)ΔT=KΔTclk,
where K is the number of inputs (K=4, in the example provided) and ΔTclk is the clock delay.Figure 14
Overview of time multiplexingK-beam filters.The signal flow inside the time-multiplexed PPCM is such that the coefficients of feedback terms ofK filters are given to a two-input multiplier through a commutating multiplexer. Input signal w(Δxn1,cΔTsn2) is given to the other input of multiplier. A counter is used to select feedback coefficient given to multiplexer. We have designed the filter bank with four filters. Hence, the counter runs from 0 to 3 and restarts. Therefore, when there exists K filter coefficients, the required counter must continuously count up 0 to log2K. The critical-path delay reduces as the architecture is fine-grain pipelined with SLA pipelining for the feedback path. Figure 15 shows each fully multiplexed multiplier circuit (FMMC) design for beam filters. 2nd-order and 3rd-order time-multiplexed designs are as shown in Figures 16 and 17, respectively, with ΔT as in (22).Figure 15
Fully multiplexed multiplier circuit forK-time-multiplexed filters.Figure 16
Time-multiplexed PPCM of 2nd-order beam filter bank with 3-stage scattered look-ahead pipelining forK-time-multiplexed beam filters.Figure 17
Time-multiplexed PPCM of 3rd-order beam filter bank with 3-stage scattered look-ahead pipelining forK-time-multiplexed beam filters.


## 4. Simulation and Implementation

### 4.1.  The 2D Frequency Response

The obtained 2D magnitude frequency response of a 2nd-order frequency-planar beam plane-wave filter matches the ideal frequency response shown in Figure18. Figure 19 shows the magnitude frequency response of 3rd-order plane-wave filter. We observe unavoidable warping effect in discrete domain systems due to the use of the 2D bilinear transform. Warping effects can be avoided by temporally oversampling the signal such that the spectrum lies totally within the straight-line region of the beam response. Figure 20 shows contours of the magnitude response of the filters in log scale. Plots for infinite precision, highest fixed-point precision, as well as lowest fixed-point precision are provided. It can be seen that the transition (roll-off) of the 3rd-order plane-wave filter is sharper compared to that of 2nd-order filter. The performance of the designed finite precision digital systolic-array circuit is measured by calculating the L2 error energy due under quantization effects. The output obtained from difference equation (from Matlab, at 64-bit precision) is considered as ideal output, and the error is calculated by the difference of ideal output and the fixed-point measurement from the FPGA device. The L2 energy of the error, Eerror, is calculated as follows:
(23)Eerror=∑n2=1N2‍∑n1=1N1‍||yideal′(ejω1,ejω2)|-|y′(ejω1,ejω2)||2,
where yideal′(ejω1,ejω2) is the Fast Fourier Transform of the ideal output obtained using difference equation from Matlab and y'(ejω1,ejω2) is the Fast Fourier Transform of the output obtained from the hardware design. We have tabulated unnormalized L2-error energies as a metric for quantization noise levels for different values of word sizes for 2nd-order and 3rd-order plane-wave filters in Tables 2 and 3, respectively.Figure 18
Magnitude frequency response of 2D 2nd-order beam filter for impulse input signal.Figure 19
Magnitude frequency response of 2D 3nd-order beam filter for impulse input signal.Contour plots of beam filter response in log scale for (a) ideal 2nd-order beam filter, (b) 2nd-order beam filter forW=27, (c) 2nd-order beam filter for W=21, (d) ideal 3rd-order beam filter, (e) 3rd-order beam filter for W=34, (f) 3rd-order beam filter for W=29.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.2. Broadband Signal Filtering

We demonstrate the directional selectivity of the proposed frequency-planar plane-wave beam filters by employing a Gaussian impulsive ultra-wideband signal [90] as input to the filter. The input to the filters that is shown in Figure 21(a) is a combination of three signals with space-time DOAs -40°, 10°, and 41° given by
(24)w(n1,n2)=cos(ωklk)e-λklk2,
where lk=n1sin(θk)+n2, k=0,1,2. We used ωk=1.12, λk=0.1, θ0, θ1, θ2 such that they have space-time DOA, ψ0=-40°, ψ1=10°, ψ2=41°. The 2nd-order beam filter selectively filters signal with space-time DOA of 10° and attenuates the signal (pulse peak) with ψ0=-40° by 40.11 dB and signal (pulse peak) with ψ2=41° by 37.22 dB. Similarly, 3rd-order beam filter selects signal with space-time DOA of 10° and attenuates the signal (pulse peak) with ψ0=-40° by 50.305 dB and signal (pulse peak) with ψ2=41° by 52.2 dB.Input and output Gaussian signals in time domain.
(a)
Input signal of three 2D Gaussian pulse functions atN1=31 with combination of three different space-time DOA signals
(b)
Output signal of 2D 2nd-order filter atN1=31
(c)
Output signal of 2D 3rd-order filter atN1=31
(d)
Cross-correlated signal of input and output signal atN1=31 for 2nd-order filter
(e)
Cross-correlated signal of input and output signal atN1=31 for 3rd-order filterThe directional attenuation of the signals was calculated using20log10|A| dB where A is the magnitude of the attenuated signal. Attenuation in signal energy was calculated by correlation. The directional enhancement of the energy of desired signal (or attenuation of the undesired signal) was calculated using 10log10|A| dB. The attenuation in energy of the signal with ψ0=-40° is 18.78 dB for 2nd-order plane-wave filter and 26.2 dB for 3rd-order plane-wave filter. Energy of the signal with ψ2=41° is attenuated by 14.26 dB by 2nd-order plane-wave filter and 26.69 dB by 3rd-order plane-wave filter. We observe that the attenuation of the undesired signals improves for the higher-order plane-wave filter, which confirms that the directional selectivity of a beamformer improves with the order of the filter. Figure 22(a) shows the 2D spectrum of the input Gaussian broadband signal. In Figure 22(b), we show the output of the 2nd-order beam filter, followed by Figure 22(c), which shows the output of 3rd-order beam filter for broadband Gaussian modulated cosine waves.Input and output Gaussian signals in frequency domain.
(a)
X-Y view of three 2D Gaussian pulse functions in frequency domain
(b)
X
-
Y view of frequency response of 2D 2nd-order filter for Gaussian pulse signals
(c)
X
-
Y view of frequency response of 2D 3rd-order filter for Gaussian pulse signals


### 4.3. Beam Patterns (Array Factor)

Directional enhancement of plane-wave beam filters can be observed through beam patterns of filter at different frequencies. Figure23 shows comparison between beam patterns of 1st-order filter, 2nd-order filter, and 3rd-order filter for frequencies π/4, π/2, 2π/3 radians, respectively. The polar plot of Nth-order filter can be obtained using magnitude of the transfer functions which is given by
(25)H(z1,z2)=(1+z1-1)N(1+z2-1)N1+∑i=0N∑j=0Nbijz1-iz2-j,
where i+j≠0 for feedback loop
(26)z1=ejω1,z2=ejω2,ω1=-ω2sinψ.
By substituting ω1, z1, and z2 in terms of ω2, H(z1,z2) is made dependent on ω2. The magnitude of normalized function H(z1,z2) is calculated by
(27)P(ψ,ω2)=|H(ejω2sinψ,ejω2)|max(|H(ejω2sinψ,ejω2)|).
The values of bij for Nth-order function are same as the values considered for filter design. The plots shown in Figure 23 give the polar plot (in dB scale) of the functions H(z1,z2) using log scale which means the magnitude of the function is calculated as 20log10|H(z1,z2)|.Beam patterns of 2D filters of (a) 3rd-order, (b) 2nd-order, (c) 1st-order along space-time DOA,ψ=10° for ω2=2π/3, ω2=π/2, ω2=π/4 in order.
(a)
(b)
(c)


### 4.4. FPGA-Based ASIC Emulation

We employ FPGA-based logic emulation for low-cost prototyping of potential RF digital application specific integrated circuits (ASICs). The proposed designs of 2nd-order and 3rd-order plane-wave beam filters are implemented on a Xilinx Virtex 6 xc6vsx475t-2ff1759 device for different fixed-point precision levels. The 3rd-order plane-wave beam filter is more complicated compared to 2nd-order filter due to higher multiplier and adder complexity. To ensure place-and-route success, we limit the number of PPCMs to 10 for each case. Implementation was first simulated and then targeted to a Virtex 6 xc6vsx475t-2ff1759 FPGA board. We optimized the real-time clock speed of the proposed designs by applying 3-stage SLA pipelining. These filters were implemented both with and without SLA in order to observe the relative improvement in performance. Tables4, 5, 6, and 7 show the FPGA hardware resources such as slice registers, look-up tables (LUTs), flip-flops (FFs), CPD, and maximum clock frequency. We observe a reduction in speed of operation of the 3rd-order plane-wave filter due to increased complexity at same level of pipelining as 2nd-order plae-wave filter. Time-multiplexed filters having corresponding folded digital architectures were physically implemented on the same board, and resource consumptions are tabulated in Tables 8 and 9, respectively. Folding and time multiplexing were applied to both designs having the SLA pipelining. At this stage, we use FPGA prototypes for verification of operation at an order of magnitude lower clock frequency than the final expected RF digital realization.Table 4
Hardware resource consumptions for pipelined 2nd-order beam filter with 3-stage SLA for different word lengths for 20 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
27
,
-
17
)
9.220
106.757
54,235
188,387
54,215
(
26
,
-
16
)
8.945
108.968
54,048
179,186
53,969
(
25
,
-
15
)
8.749
111.271
51,926
165,477
51,776
(
24
,
-
14
)
8.654
113.960
47,707
155,265
47,707
(
23
,
-
13
)
8.442
116.536
41,103
120,694
41,065
(
22
,
-
12
)
8.297
119.360
41,103
120,431
41,065Table 5
Hardware resource consumptions for pipelined 2nd-order beam filter without LA for different word lengths for 20 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
27
,
-
17
)
10.390
94.099
36,098
107,604
36,078
(
26
,
-
16
)
10.069
98.260
36,502
105,162
36,482
(
25
.
-
15
)
9.825
99.482
35,010
97,111
34,860
(
24
,
-
14
)
9.766
99.900
31,648
92,578
31,610
(
23
,
-
13
)
9.695
102.048
28,590
78,706
28,570
(
22
,
-
12
)
9.656
102.222
26,578
72,774
26,558Table 6
Hardware resource consumptions for pipelined 3rd-order beam filter with SLA for different word lengths for 10 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
34
,
-
19
)
12.703
76.417
62,330
209,945
62,263
(
33
,
-
18
)
11.717
83.458
60,435
192,506
60,327
(
32
,
-
17
)
11.048
89.365
58,581
182,595
58,509
(
31
,
-
16
)
10.751
90.9421
57,761
175,520
57,268
(
30
,
-
15
)
9.711
101.389
54,833
157,486
54,479
(
29
,
-
14
)
9.593
103.274
52,931
146,335
52,490Table 7
Hardware resource consumptions for pipelined 3nd-order beam filter without LA for different word lengths for 10 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
34
,
-
19
)
12.983
75.506
51,546
107,959
51,510
(
33
,
-
18
)
12.977
76.207
50,747
101,496
50,666
(
32
,
-
17
)
12.939
76.262
48,166
96,815
48,157
(
31
,
-
16
)
11.172
87.777
63,434
86,993
63,416
(
30
,
-
15
)
11.012
89.039
39,937
85,058
39,919
(
29
,
-
14
)
10.843
91.082
39,255
79,450
39,172Table 8
Hardware resource consumptions for time-multiplexed 2nd-order beam filter for different word lengths for 10 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
27
,
-
17
)
16.210
60.55
27,497
169,646
27,159
(
26
,
-
16
)
15.925
62.38
26,717
160,303
26,035
(
25
,
-
15
)
15.627
63.5
25,055
149,922
24,903
(
24
,
-
14
)
15.363
64.4
24,137
137,879
23,803
(
23
,
-
13
)
14.056
69.9
23,066
136,398
22,818
(
22
,
-
12
)
13.747
71.8
22,276
125,381
21,686Table 9
Hardware resource consumptions for time-multiplexed 3rd-order beam filter for different word lengths for 6 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
34
,
-
19
)
25.897
38.77
19,299
195,257
18,690
(
33
,
-
18
)
25.438
39.24
19,250
187,678
18,054
(
32
,
-
17
)
24.487
40.17
18,217
173,988
17,461
(
31
,
-
16
)
24.477
40.82
17,670
160,669
16,808
(
30
,
-
15
)
23.315
42.56
17,067
152,211
16,203
(
29
,
-
14
)
23.065
43.51
16,689
141,785
15,610


### 4.5. Computational Complexity

Table10 shows comparison of computational complexity and throughput for one PPCM between 2nd-order and 3rd-order beam filters for different stages of SLA. VLSI metrics area time (AT2n, 0≤n≤1) [91] is calculated as a measurement of complexity to find the main constraints in designing the filters. In VLSI systems, n=1 is used for cases where low chip area is more important than clock speed. Similarly, n=2 is used for cases where clock speed is the driving factor. Table 11 gives measures of both AT and AT2 for 2nd-order pipelined designs, both with and without LA, for different levels of fixed-point precision. Furthermore, Table 12 gives the same metrics for the 3rd-order filter architecture.Table 10
Computational complexity and throughput of 2nd-order and 3rd-order beam filters of N PPCMs.
N
Order of
Mults.
Mult.s
Adders
Add./sub.s
SLA
per PPCM
per sec
per PPCM
per sec
2
None
8
8
N
F
clk
8
8
N
F
clk
3
None
15
15
N
F
clk
15
15
N
F
clk
2
2
10
10
N
F
clk
10
10
N
F
clk
3
2
18
18
N
F
clk
18
18
N
F
clk
2
3
12
12
N
F
clk
12
12
N
F
clk
3
3
21
21
N
F
clk
21
21
N
F
clkTable 11
Area time constraints for 2nd-order filter.
W
Without LA
With SLA
A
T
A
T
2
A
T
A
T
2
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
27
1118.1
11616.1
1736.9
16014.5
26
1058.9
10661.8
1602.8
14337.2
25
954.1
9374.2
1447.8
12666.4
24
904.1
8829.6
1343.7
11628.1
23
763.1
7397.8
1018.9
8601.5
22
702.7
6785.3
999.2
8290.5Table 12
Area time constraints for 3rd-order filter.
W
Without LA
With SLA
A
T
A
T
2
A
T
A
T
2
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
34
1401.6
18197.4
2666.9
33878.0
33
1317.1
17092.2
2255.6
26428.8
32
1252.7
16208.5
2017.3
22287.2
31
971.9
10857.9
1887.0
20287.3
30
936.6
10314.5
1529.3
14851.4
29
861.5
9341.0
1403.7
13466.5


## 4.1.  The 2D Frequency Response

The obtained 2D magnitude frequency response of a 2nd-order frequency-planar beam plane-wave filter matches the ideal frequency response shown in Figure18. Figure 19 shows the magnitude frequency response of 3rd-order plane-wave filter. We observe unavoidable warping effect in discrete domain systems due to the use of the 2D bilinear transform. Warping effects can be avoided by temporally oversampling the signal such that the spectrum lies totally within the straight-line region of the beam response. Figure 20 shows contours of the magnitude response of the filters in log scale. Plots for infinite precision, highest fixed-point precision, as well as lowest fixed-point precision are provided. It can be seen that the transition (roll-off) of the 3rd-order plane-wave filter is sharper compared to that of 2nd-order filter. The performance of the designed finite precision digital systolic-array circuit is measured by calculating the L2 error energy due under quantization effects. The output obtained from difference equation (from Matlab, at 64-bit precision) is considered as ideal output, and the error is calculated by the difference of ideal output and the fixed-point measurement from the FPGA device. The L2 energy of the error, Eerror, is calculated as follows:
(23)Eerror=∑n2=1N2‍∑n1=1N1‍||yideal′(ejω1,ejω2)|-|y′(ejω1,ejω2)||2,
where yideal′(ejω1,ejω2) is the Fast Fourier Transform of the ideal output obtained using difference equation from Matlab and y'(ejω1,ejω2) is the Fast Fourier Transform of the output obtained from the hardware design. We have tabulated unnormalized L2-error energies as a metric for quantization noise levels for different values of word sizes for 2nd-order and 3rd-order plane-wave filters in Tables 2 and 3, respectively.Figure 18
Magnitude frequency response of 2D 2nd-order beam filter for impulse input signal.Figure 19
Magnitude frequency response of 2D 3nd-order beam filter for impulse input signal.Contour plots of beam filter response in log scale for (a) ideal 2nd-order beam filter, (b) 2nd-order beam filter forW=27, (c) 2nd-order beam filter for W=21, (d) ideal 3rd-order beam filter, (e) 3rd-order beam filter for W=34, (f) 3rd-order beam filter for W=29.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.2. Broadband Signal Filtering

We demonstrate the directional selectivity of the proposed frequency-planar plane-wave beam filters by employing a Gaussian impulsive ultra-wideband signal [90] as input to the filter. The input to the filters that is shown in Figure 21(a) is a combination of three signals with space-time DOAs -40°, 10°, and 41° given by
(24)w(n1,n2)=cos(ωklk)e-λklk2,
where lk=n1sin(θk)+n2, k=0,1,2. We used ωk=1.12, λk=0.1, θ0, θ1, θ2 such that they have space-time DOA, ψ0=-40°, ψ1=10°, ψ2=41°. The 2nd-order beam filter selectively filters signal with space-time DOA of 10° and attenuates the signal (pulse peak) with ψ0=-40° by 40.11 dB and signal (pulse peak) with ψ2=41° by 37.22 dB. Similarly, 3rd-order beam filter selects signal with space-time DOA of 10° and attenuates the signal (pulse peak) with ψ0=-40° by 50.305 dB and signal (pulse peak) with ψ2=41° by 52.2 dB.Input and output Gaussian signals in time domain.
(a)
Input signal of three 2D Gaussian pulse functions atN1=31 with combination of three different space-time DOA signals
(b)
Output signal of 2D 2nd-order filter atN1=31
(c)
Output signal of 2D 3rd-order filter atN1=31
(d)
Cross-correlated signal of input and output signal atN1=31 for 2nd-order filter
(e)
Cross-correlated signal of input and output signal atN1=31 for 3rd-order filterThe directional attenuation of the signals was calculated using20log10|A| dB where A is the magnitude of the attenuated signal. Attenuation in signal energy was calculated by correlation. The directional enhancement of the energy of desired signal (or attenuation of the undesired signal) was calculated using 10log10|A| dB. The attenuation in energy of the signal with ψ0=-40° is 18.78 dB for 2nd-order plane-wave filter and 26.2 dB for 3rd-order plane-wave filter. Energy of the signal with ψ2=41° is attenuated by 14.26 dB by 2nd-order plane-wave filter and 26.69 dB by 3rd-order plane-wave filter. We observe that the attenuation of the undesired signals improves for the higher-order plane-wave filter, which confirms that the directional selectivity of a beamformer improves with the order of the filter. Figure 22(a) shows the 2D spectrum of the input Gaussian broadband signal. In Figure 22(b), we show the output of the 2nd-order beam filter, followed by Figure 22(c), which shows the output of 3rd-order beam filter for broadband Gaussian modulated cosine waves.Input and output Gaussian signals in frequency domain.
(a)
X-Y view of three 2D Gaussian pulse functions in frequency domain
(b)
X
-
Y view of frequency response of 2D 2nd-order filter for Gaussian pulse signals
(c)
X
-
Y view of frequency response of 2D 3rd-order filter for Gaussian pulse signals


## 4.3. Beam Patterns (Array Factor)

Directional enhancement of plane-wave beam filters can be observed through beam patterns of filter at different frequencies. Figure23 shows comparison between beam patterns of 1st-order filter, 2nd-order filter, and 3rd-order filter for frequencies π/4, π/2, 2π/3 radians, respectively. The polar plot of Nth-order filter can be obtained using magnitude of the transfer functions which is given by
(25)H(z1,z2)=(1+z1-1)N(1+z2-1)N1+∑i=0N∑j=0Nbijz1-iz2-j,
where i+j≠0 for feedback loop
(26)z1=ejω1,z2=ejω2,ω1=-ω2sinψ.
By substituting ω1, z1, and z2 in terms of ω2, H(z1,z2) is made dependent on ω2. The magnitude of normalized function H(z1,z2) is calculated by
(27)P(ψ,ω2)=|H(ejω2sinψ,ejω2)|max(|H(ejω2sinψ,ejω2)|).
The values of bij for Nth-order function are same as the values considered for filter design. The plots shown in Figure 23 give the polar plot (in dB scale) of the functions H(z1,z2) using log scale which means the magnitude of the function is calculated as 20log10|H(z1,z2)|.Beam patterns of 2D filters of (a) 3rd-order, (b) 2nd-order, (c) 1st-order along space-time DOA,ψ=10° for ω2=2π/3, ω2=π/2, ω2=π/4 in order.
(a)
(b)
(c)


## 4.4. FPGA-Based ASIC Emulation

We employ FPGA-based logic emulation for low-cost prototyping of potential RF digital application specific integrated circuits (ASICs). The proposed designs of 2nd-order and 3rd-order plane-wave beam filters are implemented on a Xilinx Virtex 6 xc6vsx475t-2ff1759 device for different fixed-point precision levels. The 3rd-order plane-wave beam filter is more complicated compared to 2nd-order filter due to higher multiplier and adder complexity. To ensure place-and-route success, we limit the number of PPCMs to 10 for each case. Implementation was first simulated and then targeted to a Virtex 6 xc6vsx475t-2ff1759 FPGA board. We optimized the real-time clock speed of the proposed designs by applying 3-stage SLA pipelining. These filters were implemented both with and without SLA in order to observe the relative improvement in performance. Tables4, 5, 6, and 7 show the FPGA hardware resources such as slice registers, look-up tables (LUTs), flip-flops (FFs), CPD, and maximum clock frequency. We observe a reduction in speed of operation of the 3rd-order plane-wave filter due to increased complexity at same level of pipelining as 2nd-order plae-wave filter. Time-multiplexed filters having corresponding folded digital architectures were physically implemented on the same board, and resource consumptions are tabulated in Tables 8 and 9, respectively. Folding and time multiplexing were applied to both designs having the SLA pipelining. At this stage, we use FPGA prototypes for verification of operation at an order of magnitude lower clock frequency than the final expected RF digital realization.Table 4
Hardware resource consumptions for pipelined 2nd-order beam filter with 3-stage SLA for different word lengths for 20 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
27
,
-
17
)
9.220
106.757
54,235
188,387
54,215
(
26
,
-
16
)
8.945
108.968
54,048
179,186
53,969
(
25
,
-
15
)
8.749
111.271
51,926
165,477
51,776
(
24
,
-
14
)
8.654
113.960
47,707
155,265
47,707
(
23
,
-
13
)
8.442
116.536
41,103
120,694
41,065
(
22
,
-
12
)
8.297
119.360
41,103
120,431
41,065Table 5
Hardware resource consumptions for pipelined 2nd-order beam filter without LA for different word lengths for 20 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
27
,
-
17
)
10.390
94.099
36,098
107,604
36,078
(
26
,
-
16
)
10.069
98.260
36,502
105,162
36,482
(
25
.
-
15
)
9.825
99.482
35,010
97,111
34,860
(
24
,
-
14
)
9.766
99.900
31,648
92,578
31,610
(
23
,
-
13
)
9.695
102.048
28,590
78,706
28,570
(
22
,
-
12
)
9.656
102.222
26,578
72,774
26,558Table 6
Hardware resource consumptions for pipelined 3rd-order beam filter with SLA for different word lengths for 10 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
34
,
-
19
)
12.703
76.417
62,330
209,945
62,263
(
33
,
-
18
)
11.717
83.458
60,435
192,506
60,327
(
32
,
-
17
)
11.048
89.365
58,581
182,595
58,509
(
31
,
-
16
)
10.751
90.9421
57,761
175,520
57,268
(
30
,
-
15
)
9.711
101.389
54,833
157,486
54,479
(
29
,
-
14
)
9.593
103.274
52,931
146,335
52,490Table 7
Hardware resource consumptions for pipelined 3nd-order beam filter without LA for different word lengths for 10 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
34
,
-
19
)
12.983
75.506
51,546
107,959
51,510
(
33
,
-
18
)
12.977
76.207
50,747
101,496
50,666
(
32
,
-
17
)
12.939
76.262
48,166
96,815
48,157
(
31
,
-
16
)
11.172
87.777
63,434
86,993
63,416
(
30
,
-
15
)
11.012
89.039
39,937
85,058
39,919
(
29
,
-
14
)
10.843
91.082
39,255
79,450
39,172Table 8
Hardware resource consumptions for time-multiplexed 2nd-order beam filter for different word lengths for 10 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
27
,
-
17
)
16.210
60.55
27,497
169,646
27,159
(
26
,
-
16
)
15.925
62.38
26,717
160,303
26,035
(
25
,
-
15
)
15.627
63.5
25,055
149,922
24,903
(
24
,
-
14
)
15.363
64.4
24,137
137,879
23,803
(
23
,
-
13
)
14.056
69.9
23,066
136,398
22,818
(
22
,
-
12
)
13.747
71.8
22,276
125,381
21,686Table 9
Hardware resource consumptions for time-multiplexed 3rd-order beam filter for different word lengths for 6 PPCMs.
(W,D)
T
CPD
F
max
FPGA source consumptions
(ns)
(MHz)
Slice Reg.s
LUTs
FFs
(
34
,
-
19
)
25.897
38.77
19,299
195,257
18,690
(
33
,
-
18
)
25.438
39.24
19,250
187,678
18,054
(
32
,
-
17
)
24.487
40.17
18,217
173,988
17,461
(
31
,
-
16
)
24.477
40.82
17,670
160,669
16,808
(
30
,
-
15
)
23.315
42.56
17,067
152,211
16,203
(
29
,
-
14
)
23.065
43.51
16,689
141,785
15,610


## 4.5. Computational Complexity

Table10 shows comparison of computational complexity and throughput for one PPCM between 2nd-order and 3rd-order beam filters for different stages of SLA. VLSI metrics area time (AT2n, 0≤n≤1) [91] is calculated as a measurement of complexity to find the main constraints in designing the filters. In VLSI systems, n=1 is used for cases where low chip area is more important than clock speed. Similarly, n=2 is used for cases where clock speed is the driving factor. Table 11 gives measures of both AT and AT2 for 2nd-order pipelined designs, both with and without LA, for different levels of fixed-point precision. Furthermore, Table 12 gives the same metrics for the 3rd-order filter architecture.Table 10
Computational complexity and throughput of 2nd-order and 3rd-order beam filters of N PPCMs.
N
Order of
Mults.
Mult.s
Adders
Add./sub.s
SLA
per PPCM
per sec
per PPCM
per sec
2
None
8
8
N
F
clk
8
8
N
F
clk
3
None
15
15
N
F
clk
15
15
N
F
clk
2
2
10
10
N
F
clk
10
10
N
F
clk
3
2
18
18
N
F
clk
18
18
N
F
clk
2
3
12
12
N
F
clk
12
12
N
F
clk
3
3
21
21
N
F
clk
21
21
N
F
clkTable 11
Area time constraints for 2nd-order filter.
W
Without LA
With SLA
A
T
A
T
2
A
T
A
T
2
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
27
1118.1
11616.1
1736.9
16014.5
26
1058.9
10661.8
1602.8
14337.2
25
954.1
9374.2
1447.8
12666.4
24
904.1
8829.6
1343.7
11628.1
23
763.1
7397.8
1018.9
8601.5
22
702.7
6785.3
999.2
8290.5Table 12
Area time constraints for 3rd-order filter.
W
Without LA
With SLA
A
T
A
T
2
A
T
A
T
2
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
[
L
]
2
[
T
]
×
1
0
-
12
[
L
]
2
[
T
]
2
×
1
0
-
21
34
1401.6
18197.4
2666.9
33878.0
33
1317.1
17092.2
2255.6
26428.8
32
1252.7
16208.5
2017.3
22287.2
31
971.9
10857.9
1887.0
20287.3
30
936.6
10314.5
1529.3
14851.4
29
861.5
9341.0
1403.7
13466.5


## 5. Conclusion

Highly directional steerable ultra-wideband digital antenna aperture arrays have useful applications in wireless communications, radar, radio astronomy, spectrum sensing, RF imaging and remote sensing. We propose novel massively-parallel systolic-array architectures for the real-time digital realization of beamforming 2D IIR frequency-planar beam plane-wave filters based on resistively terminated LC ladder networks. The proposed architectures are aimed at 2nd- and 3rd-order beam filters and are based on the recently proposed 1st-order beam filter architecture [10]. These beamformers offer both low computational complexity and UWB performance.The proposed architectures enable RF throughputs when eventually realized using high-speed CMOS technology. The proposed architectures are evaluated for correct operation, area, time, and complexity metrics as well as beam sensitivity and noise levels as a function of finite precision. Extensive prototype FPGA realizations, simulations, and FPGA-based emulations of beamforming performance are used here to validate the architectures and design procedures for high-performance digital UWB beamforming applications. As part of the proposed optimized designs, a fine-grain pipelined systolic-array 2D IIR plane-wave beam filters of orders 2 and 3 have been designed and optimized using scattered look-ahead to reduce the CPD of the digital circuits. Low CPD results in high real-time throughput with corresponding increase in clock frequency. The architectures were evaluated using CPD (T), area (A), as well as AT and AT2.The proposed 2nd-order design filter architecture is verified using a realistic Gaussian modulated cosine wave input signal consisting of several plane-waves. The filter is designed to enhance the plane-wave having space-timeDOA=10°, while attenuating the undesired signal up to 40.11 dB for space-time DOA=-40° and 37.22 dB for space-time DOA=41°. Similarly, the 3rd-order filter provided up to 50.305 dB of stopband rejection for space-time DOA=-40° and 52.2 dB of stop-band rejection for space-time DOA=41° for a broadband Gaussian-modulated cosine-wave signal. The corresponding hardware architectures were verified on FPGA chip using measured unit impulse responses obtained using stepped FPGA hardware cosimulation. The 2D frequency response of the measured impulse response from the FPGA chip is compared with a reference response obtained from the difference equation by calculating the L2 error energy for different finite precision sizes.Furthermore, the proposed architectures were folded and time-multiplexed to form a filter bank. The time-multiplexed filters have also been physically implemented on Xilinx Virtex6 xc6vsx475t-2ff1759 FPGA board. Future work involves digital CMOS integrated circuits using ASIC standard cells for RF clock frequencies with applications in UWB real-time electromagnetic beamforming antenna arrays.



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*Source: 234263-2012-09-17.xml*