Discrete-Time Signals and Systems

Transcription

1 ECE 46 Lec Viewgraph of 35 Discrete-Time Signals and Systems Sequences: x { x[ n] }, < n<, n integer. Sampling: xn [ ] x a ( nt), < n < () 0, n 0,, n 0, Important Sequences: unit sample δ[ n], unit step un [ ] (), n 0. 0, n < 0. Show that: un [ ] δ[ k], un [ ] δ[ n k], and k k 0 δ[ n] un [ ] un [ ] Also, the sifting property, xn [ ] xk [ ]δ[ n k] (4) k xn [ ] is a linear combination of appropriately delayed unit samples. Exponential and sinusoidal sequences: xn [ ] Aα n, xn [ ] Ae jω 0n, xn [ ] Acos( ω 0 n + φ) n (3) Measure of a sequence, l p : Energy of a sequence: xn [ ] p xn [ ] p n ξ x p (5). Similar to xt () xτ ()δt ( τ)dτ in continuous-time systems.

2 ECE 46 Lec Viewgraph of 35 Linear Discrete-Time Systems: x[n] T{.} y[n] yn [ ] T{ x[ n] } (6) Linear Systems: Linear Superposition ( additivity + scaling or homogeneity) T{ ax [ n] + bx [ n] } at{ x [ n] } + bt{ x [ n] } (7) From xn [ ] xk [ ]δ[ n k] (4), and linearity (7): k yn [ ] T{ x[ n] } T x[ k]δ[ n k] (8) k xk [ ]T{ δ[ n k] } k Let h k [ n] T{ δ[ n k] } (9) Then yn [ ] xk [ ]h k [ n] (0) k Causality: If x [ n] x [ n] for n n 0, then x [ n] x [ n] for n n 0. Stability: BIBO iff every bounded input produces a bounded output summable) h k [ n] k, finite (bounded,

3 ECE 46 Lec Viewgraph 3 of 35 Linear Time-Invariant Systems (LTI): h k [ n] hn [ k], hn [ ] h 0 [ n] () Since T{ δ[ n k] } hn [ k], then yn [ ] xk [ ]hn [ k] xn [ ] () k hn [ ] y is the (discrete-time) convolution of x with h. Convolution Sum Show that convolution is commutative: Other Properties: yn [ ] hk [ ]xn [ k] hn [ ] k xn [ ] xn [ ] hn [ ] Parallel Combination (Connection): hn [ ] h [ n] + h [ n], H H + H Cascade Connection: hn [ ] h h h h, H H H

4 ECE 46 Lec Viewgraph 4 of 35 Frequency Domain Representation of Discrete-Time Signals and Systems (Frequency Response) Let xn [ ] e jωn, < n <, then yn [ ] hk [ ]xn [ k] k hk [ ]e jω( n k) e jωn hk [ ]e jωk k k (3) Define He jω ( ) hk [ ]e jωk k (4) then yn [ ] He ( jω )e jωn (5) e jωn is an eigenfunction of the system. H( e jω ) is its associated eigenvalue, and is called the frequency response of the system. H H R + jh I He j H H is periodic in ω with period π, and hence has a Fourier series representation (7).Therefore hn [ ] π He ( jω )e jωn dω π π Representation of Sequences by Fourier Transforms: Any sequence, with suitable convergence conditions: Inverse Fourier Transform xn [ ] π Xe ( jω )e jωn dω, (6) π π Fourier Transform Xe jω ( ) xk [ ]e jωk (7) k

5 ECE 46 Lec Viewgraph 5 of 35 Frequency Response: From convolution (5), yn [ ] xk [ ]hn [ k], Fourier transform of output is k Ye ( jω ) yn [ ]e jωn hn [ k]xk [ ]e jωn n xk [ ] hn [ k]e jωn n k k n xk [ ]e jωk hn [ k]e jω( n k) n Symmetry Properties of the Fourier Transform: k Xe jω ( )He jω ( ) (8) Let xn [ ] x e [ n] + x o [ n], where x e [ n] -- ( xn [ ] + x [ n] ), x o [ n] -- ( xn [ ] x [ n] ) (9) Fourier Transform Theorems: TABLE. Fourier Transform Theorems Sequence Theorem Fourier Transform ax + by Linearity ax + by xn [ n d ] Time Shift e jωn d X( e jω ) e jω 0n jω ( ω xn [ ] Frequency Shift Xe 0 ) ( ) x[ n] Time Reversal X( e jω ), if real x then X ( e jω )

6 ECE 46 Lec Viewgraph 6 of 35 d jω nx[ n] Differentiation in Frequency j Xe ( ) dω x y xy Convolution Modulation or Windowing Parsevals Theorem π XY Xe ( jθ )Ye ( jω ( θ) )dθ π π n xn [ ] π Xe ( jω ) dω, π xn [ ]y [ n] π n π Xe ( jω )Y ( e jω )dω π π

7 ECE 46 Lec Viewgraph 7 of 35 Linear Constant-Coefficient Difference Equations th-order linear constant-coefficient difference equation M M a k yn [ k ] a k b r b r xn [ r], yn [ ] yn [ k] + (0) a xn [ r] a 0 k 0 r 0 k r 0 Let a 0. Causality, initial conditions, stability, IIR and FIR filters... If a i 0, i,,,, then FIR filter, otherwise IIR. Examples of Filters: --- Averaging Filter yn [ ] --- xn [ r], () hn [ ], 0 n, r 0 0, else He jω ( ) hn [ ]e jωn --- jωn e --- e j ω sin ω j ( ) ω () n n 0 e j ω e sin ω --- Ideal Lowpass filter H e j ω, ω ω c, ω ( ), hn [ ] c e j ω n sinω c n d ω (3) 0, ω c < ω < π π ω c πn Digital Interpolator He ( jω ) e jωτ hn [ ]e jωn, 0 < τ<. Since hn [ ] π He ( jω )e jωn dω, then π π hn [ ] sinπ( n τ) sinπτ π( n τ) π ( ( n τ) ) n+ sinπτ πτ ( ( n τ ) ) n + (4)

8 ECE 46 Lec Viewgraph 8 of 35 Periodic (uniform) Sampling: Sampling of Continuous-Time Signals x[ n] x c ( nt), < n < (5) Sampling Frequency: T (6) f s Let x[ n] c () t X c ( jω)ejωt dω, π π Xe ( jω )e jωn dω, π xn [ ] x c ( nt) X c ( jω)e jωnt dω π T X π c ( j Ω + j π r )e jωnt e jπrn dω r π T T ω Ω ---, then T xn [ ] π π π X e jω ( ) X c ( jω) x c ()e t jωt d Xe ( jω ) xk [ ]e jωk k r (7) (8) ( r + ) ( π T) X c ( jω)e jωnt dω (9) ( r ) ( π T) π T X π c ( j Ω + j π r ) e jωnt dω π T r T j ω -- X T c ( j π r ) e jωn dω r T T j ω -- X T c ( j π r ) r T T (30) (3) (3) yquist Sampling Theorem: If x c () t is bandlimited with X c ( jω) 0 for Ω > Ω, then x c () t is uniquely determined by its samples x[ n] x c ( nt), n 0, ±, ±,, if Ω s ( π) T > Ω.

9 ECE 46 Lec Viewgraph 9 of 35 Reconstruction of a Bandlimited Signal from its Samples: x r () t xn [ ] sinc( t nt ), T sincx () sinπx πx If xn [ ] x c ( nt), and X c ( jω) 0 for Ω > Ω, then x r () t x c () t. (34) Interpolation,Digital Interpolator He ( jω ) e jωτ hn [ ]e jωn, 0 < τ < Since hn [ ] π He ( jω )e jωn dω, then π π (33) hn [ ] sinπ( n τ) sinπτ π( n τ) π ( ( n τ) ) n+ sinπτ πτ ( ( n τ ) ) n + (35) on-ideal Sampling: Example, averaging window: x [ n] -- nt τ x ()dt ( nt τ) t c (36) τ, 0 t< τ, Let r τ () t (37) 0 else Then x [ n] r τ ( nt ζ)x c ()dζ ζ (38) t Let x ct () rt ( ζ )x ()dζ ζ,, then (39) c X c ( jω) R τ ( jω)x c ( jω)

10 ECE 46 Lec Viewgraph 0 of 35 Ωτ sin j Ωτ where R τ ( jω) , Group delay (40) Ωτ e τ Discrete-Time Processing of Continuous-Time Signals: If X c ( j Ω) is bandlimited, and the sampling rate is above the yquist rate, then ( jω) H eff ( jω)x c ( jω), H eff ( jω) He ( jω ), Ω < π T, 0, Ω π T (4) Examples of Filters: Ideal Lowpass filter H e jω ( ), ω ω c, 0, ω c < ω < π (4) ω hn [ ] c e j ω n sinω c n (43) π d ω ω c πn If input is samples of bandlimited signal, sampled above yquist rate, then after ideal D/A, system behaves, ΩT < ω c, or, Ω < ω c T as H eff ( jω) (44) 0, ΩT > ω c, or, Ω > ω c T Ideal Bandlimited Differentiator

11 ECE 46 Lec Viewgraph of 35 Impulse Invariance: Given H c ( jω) which is bandlimited, then to choose H( e jω ) such that H eff ( jω) H c ( jω) : H e jω jω ( ) H c ( ), ω < π, (45) T with the further requirement that T is chosen such that H c ( jω) 0, Ω π T. Then, hn [ ] Th c ( nt), or impulse invariance. If H c ( jω) is not bandlimited, then H( e jω jω ) H c ( j π r ) r T T (46) Frequency Scaling:

12 ECE 46 Lec Viewgraph of 35 The z-transform Xz () xn [ ]z n (47) n Let z re jω, Xre jω ( ) ( xn [ ]r n )e jωn (48) n This is the Fourier transform of xn [ ]r n. Uniform convergence requires absolute summability. For n xn [ ]r n <, this happens for R- < z < R+. Replacing z by e jω in Xz () results in the Fourier transform of xn [ ], provided the unit circle is in the region of convergence of the z-transform. Uniform convergence of the z-transform, corresponds to uniform convergence of the corresponding Fourier transform, for sequences defined for a range of r s. either of x [ n] sinω c n , x πn [ n ] cosω 0 n, < n < (49)

13 ECE 46 Lec Viewgraph 3 of 35 is absolutely summable for any r. Thus no z-transform. But x has finite energy, for which the Fourier transform converges in the mean-square sense to a discontinuous periodic function. The sequence x is neither absolutely nor square summable, but a Fourier transform using impulses is possible. The Fourier transform is not continuous, infinitely differentiable functions in both cases, so they cannot result from evaluating a z-transform on the unit circle, but we use a notion that implies this. The z-transform is particularly useful when expressed in closed form, as in the case of rational functions Xz () Pz () Qz () (50) Examples: exponential and unit step: xn [ ] a n un [ ] Xz () a n un [ ]z n, n ( az ) n n 0 Convergence requires az n <, which is true for az <,or z > a. n 0 In region of convergence z Xz () az , z a z > a The z- transform converges for any finite value of a, the Fourier transform only for a <. (5) (5) For a, x[n] is unit step sequence with z-transform Xz () z, z > (53)

14 ECE 46 Lec Viewgraph 4 of 35 Let Digital Filter Structures + yn [ ] a k yn [ k k ] M b r xn [ r] r 0 Hz () M b k z k k a k z k k Yz () Xz () (54) (55) State Space Representation Let M (discussion), express in powers of z, and isolate direct coupling: β k z k k 0 H() z d z + α k z k k Let state variable be the vector v() n of dimension. Then state space representation vn ( + ) Av() n + bx() n, yn ( ) cv() n + dx() n (56) (57)

15 ECE 46 Lec Viewgraph 5 of , b, c β β β 0 α α α 0 Hz () czi ( A) b + d (58) (59) h[ n] ca n b, n,, d, n 0 (60) on-singular Transformations Input and output invariant, but Tv( n + ) TAT Tv() n + Tbx() n, yn () ct Tv() n + dx() n (6) v By changing T, new structures are obtained that have different computational, roundoff, and coefficient sensitivity properties. Changing T results in changing the basis in the state vector space in which the system is represented. Tv à TAT c ct b Tb (6)

16 ECE 46 Lec Viewgraph 6 of 35

17 ECE 46 Lec Viewgraph 7 of 35 The Discrete Fourier Transform xn [ ] j π nk --- Xk ()e, k 0 j π nk Xk () xn [ ]e, n 0 (63) (64) z plane π/. W Finite and Periodic Sequences: W j π e

18 ECE 46 Lec Viewgraph 8 of 35 Xz () xn [ ]z n n 0 Xe ( jω ) xn [ ]e jωn π, ω k, k 0,,,. n 0 j π k Xk () Xe ( ) j π k Xk () Xe ( ) xn [ ] j π nk xn [ ]e, k 0,,, n 0 nk xn [ ]W, k 0,,, n Xk ()W nk, n 0,,,. k 0 (65) (66) (67) (68) (69) Constructing Xz () from Frequency Samples for Finite Sequences: Xz () xn [ ]z n n Xk ()W nk n 0 k 0 n z (70)

19 ECE 46 Lec Viewgraph 9 of 35 k --- Xk () ( W z ) n z --- Xk () k k 0 n 0 k 0 W z Infinite Sequences: z Xk () k k 0 W z (7) Let Xz () xn [ ]z n n (7) X () k Xz () z e j π k kn xn [ ]W, k 0,, n (73) The inverse DFT of X () k is x n [ ], defined by x [ n] --- X ()W k nk, resulting in, n 0,,,. k 0 xn [ ] --- Xk ()W nk km kn --- xm [ ]W W xm [ ] --- W kn ( m) k 0 k 0m m k 0 (74) Using then kn ( m) --- W k 0 m, n + r 0, else, as shown ks --- W k 0 for s r --- W ks ; k 0, s r 0, otherwise --- W s s 0 W

20 ECE 46 Lec Viewgraph 0 of 35 So, x is an aliased version of x. Properties of the DFS/DFT: x [ n] xn [ + r] r (75) Linearity: x 3 [ n] ax [ n] + bx [ n] X 3 () k ax () k + bx () k. Both x and x are periodic of same period, or finite of same length. Circular Shift: x [ n] x [ n + m] X () k W km X k () Circular (Periodic) Convolution: x 3n [ ] x [ m]x [ n m] X 3 () k m 0 X ()X k () k x 3 [ n] x [ n]x [ n] X 3 () k --- X ()X r ( k r) r 0 (76) (77) Parseval s Relation for the DFT: n 0 x [ n] Linear Convolution Using the DFT: --- X () k k 0 (78)

21 ECE 46 Lec Viewgraph of 35 Let x [ n] be of length L, x [ n] of length M. Then x 3 x x is of length L + M. Pad each of x [ n] and x [ n] with zeros so that each is of length L + M. Compute DFT s, multiply point wise, IDFT and keep only L + M. If one of the sequences is very long, it is decomposed into sections of appropriate size, convolved, and the partial results combined according to one of the following two schemes: Overlap-add: Overlap-save:

22 ECE 46 Lec Viewgraph of 35 Divide and Conquer Algorithms: Matrix Multiplication: Computation of the DFT Strassen Algorithm: A B C, all n n. Let n be even for the next step, power of two n s for the complete algorithm. n n The first step is to divide each matrix into four matrices as follows. A A B B C C (79) A A B B C C n n Instead of the eight matrix multiplications, and four matrix additions needed in direct computation of the entries of C, the following seven products are computed. P ( A + A )( B + B ) P ( A + A )B P 3 A ( B B ) P 4 A ( B B ) P 5 ( A + A )B P 6 ( A A )( B + B ) P 7 ( A A )( B + B ) (80) Then C is computed in terms of additions of the P s.

23 ECE 46 Lec Viewgraph 3 of 35 C A B + A B P + P 4 P 5 + P 7 C A B + A B P + P 4 C A B + A B P 3 + P 5 C A B + A B P + P 3 P + P 6 (8) n n Instead of 8 multiplications and 4 additions of -- --, the algorithm requires 7 multiplications, and8 n n additions of matrices. Since multiplication of m m matrices requires m 3 scalar multiplications, and m 3 m scalar additions, removing one matrix multiplications at the cost of adding several matrix additions results in computation reduction for sufficiently large m. So far we have applied divide and conquer only once. If the rest of the computation of the submatrices is done by direct approach, and if we count a scalar multiply as approximately as costly as a scalar add, as is the case in floating point arithmetic, then the break even point is for n 30, and the savings approach /8 of the regular approach as n increases. But we can apply the algorithm again and again to all the resulting submatrices as long as this results in savings. If the algorithm is applied repeatedly until computation is completed, then the number of multiplications and additions needed to compute the multiplication of r r matrices, M() r and Ar () respectively, satisfy which has the solution, for n s (8) Ms () 7 s log 7 ( ) s log 7 n n.807, As () 67 ( s 4 s log 7 ) 6( n n ) (83)

24 ECE 46 Lec Viewgraph 4 of 35 Mr ( + ) 7Mr (), M0 (), Ar ( + ) 7Ar ( ) r, A() 0 0 Polynomial Multiplication: Let P n and Q n be polynomials of degree n- each, where n s. P n ()Q x n () x ( p 0 + p x + p x + + p n x n )( q 0 + q x + q x + + q n x n ) n n n p 0 p x p x p n x x p -- n p n x p n x p n x (84) n n n q 0 q x q x q n x x q -- n q n x q n x q n x n n P n + x P n Q --, --, n + x Q n --, --,

25 ECE 46 Lec Viewgraph 5 of 35 P n Q n --, --, n -- + x P n Q n + P n Q n + --, --, --, --, x n P n Q n --, --, ow the trick is to compute P n --, Q n, P n Q n, Q n Q n, P n P n --, --, --, --, --, --, --,, and recognize that P n Q n --, --, + P n Q n Q n Q n Pn --, --, --, --, --, P n + P n Q n + --, --, --, P n Q n --, --, n So, direct computation results in 4 multiplications and one additions of polynomials of degree --. With log 3 algorithm, 3 multiplications and 4 additions. Repeated application results in 3n n.58 multiplications, log 3 and 4n additions. Decimation-in-Time FFT: j π k X() k Xe ( ) nk xn [ ]W, k 0,,, n 0 (85)

26 ECE 46 Lec Viewgraph 6 of 35 Let γ j π -----, and notice that W e implies W r, W, W j, W j, W W r. Xk () nk xn [ ]W + nk xn [ ]W n even n odd Xk () xr [ ]( W ) rk k W x[ r + ] ( W ) rk +, W W r 0 r 0 (86) (87) k. X() k Gk () + W Hk (), k 0,,, (88) / point DFT s One point DFT is replaced by two, / point DFT s at the cost of multiplications, and additions. So the number of either additions or multiplications for a DFT of length i satisfies Let C( i ) A i, then Which has the solution C( i ) C( i ) + i A i A i + i, A 0 0 A γ γ A 0 + γ γ log (89) (90) (9)

27 ECE 46 Lec Viewgraph 7 of 35 Multidimensional Signals and DFT: Impulse response hk [, k, k, n ]. -Dimensional convolution:. (9) yk [, k, k, n ] hk [ l, k l, k, n l n ]xl [, l, l, n ] l l l n (93) y( k) x( k) * h( k) Y( z, z,, z n ) Xz (, z, z, n )Hz (, z, z, n ) k k k n Xz (, z, z, n ) xk [, k, k, n ]z z zn k k k n X e jω e jω e jω n (,,, ) xk [, k, k, n ]e j ( ω k + ω k + + ω n k n ) k k k n (94) (95) (96) n-dimensional DFT of finite length signal is sampling of n-d F.T. Many -D DFT s. n Xl (, l, l, n ) xk [, k, k, n ]e k 0k 0 k n 0 jπ l k l k l nk n n (97) -D FFT Derivation in Multidimensional Framework:

28 ECE 46 Lec Viewgraph 8 of 35 j π k Xk () Xe ( ) j π nk xn [ ]e, k 0,,, n 0 Let LM, and use k ( l + Ll ); l 0,,, L ; l 0 M,,,., then (98) ow define the LxM arrays to: M Xl ( + Ll ) e k 0 j----- π k M l e j π k l L k 0 xmk [ + k ]e j π k L l x [ k, k ] xmk [ + k ]; X ( l, l ) Xl ( + Ll ), then the -D DFT is equivalent - M, L-point DFT s along st dimension of x [ k, k ]. j π l k - Multiplication of result of step by e, l 0 L,,, k 0 M,,,. 3- L, M-point DFT s along nd dimension to finally obtain X ( l, l ). n For n, the cost is C ( ) ( i )C ( i ) + ( n ). For radix- FFT, this i becomes C( n ) n n C() + ( n ) n C() log log log (99)

29 ECE 46 Lec Viewgraph 9 of 35 Filter Design Techniques Design of Discrete-Time IIR Filters from Continuous-Time Filters: Bilinear Transformarion: s z z z s s -- Ω tan( ω ) (00) (0) (0) Imaginary axis in s-plane maps onto unit circle in z-plane, left half plane onto unit disk. Due to nonlinearity of map: - only for filters with piecewise constant magnitude response. - phase response is not critical. Given the specifications of the desired Hz (), the specifications are mapped onto those of H c () s, which is then designed as a continuous-time filter, and then transformed via bilinear transformation into desired digital filter. Butterworth Filters: Chebyshev Filters:

30 ECE 46 Lec Viewgraph 30 of 35 Elliptic Filters: Chebyshev Polynomials T n () x for n 40,,. T n () x cos( ncos x) T 0 () x T, () x x, T () x x, T 3 () x 4x 3 3x, T 4 () x 8x 4 8x + (03) (04) Relative Order of Digital Filters: Let δ δ δ, then 0log 0 δ δ 5 FIR F 0logδ 7.95 log e δ FIR F ω (05) (06)

31 ECE 46 Lec Viewgraph 3 of 35 log e δ n B ω log e δ n C ω n E 0.3log e δ ω log e (07) (08) (09) Types of Filters: Minimum phase, Linear phase, and all pass. section 5.4 to end of Ch. 5. Powers of z: Effect on impulse, frequency, and pole-zero locations. use in filter design and multirate filtering. umerical Design Methods: IIR Filters: Deczky s Method: section 7.3 FIR Filters: Windowing: section 7.4, 7.5. Optimum Aproximation: section 7.6. Best mean-square error is just a rectangular-windowed version of desired impulse response. Best min-max design (equiripple) via Parks-Mclellan algorithms and the Remez exchange method.

32 ECE 46 Lec Viewgraph 3 of 35 Computationally Efficient Digital Filter Structures Structures Based on Periodicities and z-powers: Periodicities: MFIR Filters: Strutures Based on IIR Filters (made to behave as FIR): MFIR: Switching and Resetting: Interpolation:

33 ECE 46 Lec Viewgraph 33 of 35 x[0] x[] x[4] x[6] / G(0) G() G() G(3) W 0 W.... X(0) X() X() X(3) x[] x[3] x[5] x[7] / H(0) H() H() H(3) W 7. X(4) X(5) X(6) X(7)

34 ECE 46 Lec Viewgraph 34 of 35 Computationally Efficient Digital Filter Structures (contd.) Discussion of papers: [] T. Saramaki, and A. Fam, Subfilter Approach for Designing Computationally Efficient FIR Filters, ISCAS-88, Espoo, Finland, pp , June 7-9, 988. [] Adly T. Fam, MFIR Filters: Properties and Applications, IEEE Trans. Acoust., Speech, Signal Processing, vol ASSP-9, pp. 8-36, Dec. 98. [3] Zhongqi Jing and Adly T. Fam, A ew Structure for arrow Transition Band, Lowpass Digital Filter Design, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-3, pp , Apr [4] A. Fam, FIR Filters that Approach IIR Filters in their Computational Efficiency, Twenty-first Annual Asilomar Conference on Signals, Systems and Computers, Pacific Grove, California, pp. 8-30, ov. -4, 987. [5] Tapio Saramäki and Adly T. Fam, Properties and Structure of Linear-Phase FIR Filters Based on Switching and Resetting of IIR Filters, ISCAS 90, ew Orleans, Louisiana, pp , May -3, 990. [6] Chimin Tsai and Adly T. Fam, Efficient Linear-Phase Filters Based on Switching and Time Reversal, ISCAS 90, ew Orleans, Louisiana, pp. 6-64, May -3, 990.

35 ECE 46 Lec Viewgraph 35 of 35 Optimal Partitioning and Redundancy Removal in Computing Partial Sums Discussion of concept as introduced in [] Adly T. Fam, Optimal Partitioning and Redundancy Removal in Computing Partial Sums IEEE Trans. Comput., vol. C-36, pp , Oct and related papers, and its applications to FIR filters as in [] Adly T. Fam, Space-Time Duality in Digital Filter Structures, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-3, pp , June 983. [3] A. Fam, A Multi-Signal Bus Architecture for FIR Filters with Single Bit Coefficients, ICASSP-84, San Diego, CA, pp , March 9-, 984.