Transform Representations • z-Transform ↔ x n [ ] X z ( ) ∞ ∑ = − Infinite power series in z -1 , with n X z ( ) x n z [ ] x [ n ] as coefficients of term in z - n =−∞ n 1 ∫ − = n 1 [ ] ( ) • direct evaluation using residue theorem x n X z z dz π 2 j C ( 留数定理 ) • partial fraction expansion ( 部分分式展 开 )of X ( z ) • long division ( 长除法 ) • power series expansion( 幂级数展开 ) 33
Transform Representations • X ( z ) converges (is finite) only for certain values of z – Sufficient condition for convergence ∞ ∑ − < ∞ n [ ] x n z =−∞ n • region of convergence < < R z R 1 2 34
Examples of Converge Regions = δ − 1. Delayed impulse x n [ ] [ n n ] 0 > > < ∞ < ∀ = z − = n converges for z 0, n 0; z , n 0; z n , 0 X z ( ) 0 0 0 0 = − − 2. Box pulse x n [ ] u n [ ] u n [ N ] − − − N N 1 1 z ∑ = − = < < ∞ n converges for 0 X z ( ) z z − − 1 1 z = n 0 < < ∞ all finite length sequences converge in the region 0 z = < 3. n [ ] [ ] ( 1) x n a u n a ∞ 1 ∑ = − = − > n n X z ( ) a z converges for z a − 1 1 az = n 0 all infinite duration sequences which are non-zero for n ≥ 0 > converge in the region z R 1 35
Examples of Converge Regions = − − − n 4. x n [ ] b u [ n 1] − 1 1 ∑ < = − − = − converges for z b n n X z ( ) b z − 1 1 bz =−∞ n all infinite duration sequences which are non-zero for n <0 < converge in the region z R 2 x [ n ] non-zero for - ∞ < n < ∞ can be viewed as a combination of 5. < < 3 and 4, giving a convergence region of the form R z R 1 2 > sub-sequence for n ≥ 0 ⇒ z R – 1 < sub-sequence for n <0 ⇒ – z R 2 < < total sequence ⇒ – R z R 1 2 36
Examples If x [ n ] has z-transform X ( z ) with ROC of r i <| z |< r o , find the z- transform, Y ( z ), and the region of convergence for the sequence y [ n ]= a n x [ n ] in terms of X ( z ) ∞ ∑ = − Solution n X z ( ) x n z [ ] =−∞ n ∞ ∞ ∑ ∑ − − = = n n n Y z ( ) y n z [ ] a x n z [ ] =−∞ =−∞ n n ∞ ∑ − = = n x n z a [ ]( / ) X z a ( / ) =−∞ n < < ROC: a r z a r i o 37
Examples The sequence x [ n ] has z-transform X ( z ). Show that the sequence nx [ n ] has z-transform – z d X ( z )/ d z Solution ∞ ∑ = − n X z ( ) x n z [ ] =−∞ n ∞ ∞ dX z ( ) 1 ∑ ∑ = − − − = − − n 1 n nx n z [ ] nx n z [ ] dz z =−∞ =−∞ n n 1 = − Z nx n ( [ ]) z 38
Inverse z-Transform 1 ∫ = − n 1 x n [ ] X z z ( ) dz π 2 j C where C is a closed contour that encircles the origin of the z- plane and lies inside the region of convergence for X ( z ) rational( 有理 ), can use a partial fraction expansion ( 部分分式展开 ) for finding inverse transforms 39
Partial Fraction Expansion 40
Example of Partial Fractions 41
Transform Properties + + Linearity ax n [ ] bx n [ ] aX ( ) z bX ( ) z 1 2 1 2 − − n Shift [ ] x n n ( ) z X z 0 0 Exponential − n 1 a x n [ ] X a z ( ) Weighting − nx n [ ] d ( ) / d Linear Weighting z X z z − X z − 1 x [ n ] Time Reversal ( ) ∗ Convolution X z H z ( ) ( ) x n [ ] h n [ ] Multiplication of 1 ∫ − 1 C X v W z v v dz ( ) ( / ) x n w n [ ] [ ] π Sequences 2 j 42
Discrete-Time Fourier Transform (DTFT) 43
Discrete-Time Fourier Transform • evaluation of X ( z ) on the unit circle in the z-plane ∞ ∑ ω − ω = = j j n ( ) ( ) [ ] X e X z x n e ω = j z e =−∞ n π 1 ∫ = ω ω ω j j n x n [ ] X e ( ) e d π − π 2 • sufficient condition for existence of Fourier transform is ∞ ∞ ∑ ∑ − = < ∞ = n [ ] [ ] , since 1 x n z x n z =−∞ =−∞ n n 44
Simple DTFTs Impulse Delayed impulse Step function Rectangular window Exponential Backward exponential 45
DTFT Examples 46
DTFT Examples X e ω − π < ω < π j Within interval is comprised of a pair of , ( ) ± ω impulses at 0 47
DTFT Examples 48
DTFT Examples 49
Fourier Transform Properties • Periodicity in ω ω ω + π = ( 2 ) j j k X e ( ) X e ( ) • Period of 2 π corresponds to once around unit circle in the z- plane • normalized frequency: f, 0 → 0.5 → 1 (independent of Fs) • normalized radian frequency: ω , 0 →π → 2 π (independent of Fs) • digital frequency : f D =f*Fs, 0 → 0.5Fs → Fs • digital radian frequency : ω D = ω *Fs, 0 →π Fs → 2 π Fs 50
Discrete Fourier Transform (DFT) 51
Discrete-Time Fourier Series • consider a periodic signal with period N (samples) = + − ∞ < < ∞ x n [ ] x n [ N ], n can be represented exactly by a discrete sum of x n [ ] sinusoids − N 1 ∑ − π = j 2 kn N / X k [ ] x n e [ ] • N Fourier series coefficients = n 0 − N 1 1 ∑ • N -sequence values π = j 2 kn N / x n [ ] X k e [ ] N = k 0 52
Finite Length Sequences • consider a finite length (but not periodic) sequence, x [ n ], that is zero outside the interval 0 ≤ n ≤ N -1 − N 1 = ∑ − n X z ( ) x n z [ ] = n 0 • evaluate X ( z ) at equally spaced points on the unit circle, π = = − j 2 k N / z e , k 0,1,..., N 1 k − 1 N ∑ = π = − π = − j 2 k N / j 2 kn N / X k [ ] X e ( ) x n e [ ] , k 0,1,..., N 1 = n 0 – looks like discrete-time Fourier series of periodic sequence 53
Relation to Periodic Sequence • consider a periodic sequence, , consisting of an infinite x n [ ] sequence of replicas of [ ] x n ∞ ∑ = + x n [ ] x n [ rN ] =−∞ r • The Fourier coefficients, , are then identical to the values X k [ ] π of for the finite duration sequence ⇒ j 2 k N / X e ( ) a sequence of N length can be exactly represented by a DFT representation of the form − N 1 ∑ − π = ≤ ≤ − j 2 kn N / X k [ ] x n e [ ] , 0 k N 1 = n 0 − N 1 1 ∑ = π ≤ ≤ − j 2 kn N / x n [ ] X k e [ ] , 0 n N 1 N = 54 n 0
Periodic and Finite Length Sequences 55
Sampling in Frequency (Time Domain Aliasing) 56
Sampling in Frequency (Time Domain Aliasing) 57
Time Domain Aliasing Example 58
DFT Properties Periodic Sequence Finite Sequence Period = N Length = N Sequence defined for all n Sequence defined for n = 0,1,…, N -1 DTFT defined for all ω DFS defined for k = 0,1,…, N -1 • when using DFT representation, all sequences behave as if they were infinitely periodic ⇒ DFT is really the representation of ∑ ∞ = + the extended periodic function [ ] [ ] x n x n rN −∞ 59
DFT Properties for Finite Sequences • X [ k ], the DFT of the finite sequence x [ n ], can be viewed as a sampled version of the z-transform (or Fourier transform) of the finite sequence (used to design finite length filters via frequency sampling method) • the DFT has properties very similar to those of the z-transform and the Fourier transform • the N values of X [ k ] can be computed very efficiently (time proportional to N log N ) using the set of FFT methods • DFT used in computing spectral estimates, correlation functions, and in implementing digital filters via convolutional methods 60
DFT Properties N-point sequences N-point DFT + + Linearity ax n [ ] bx n [ ] aX k [ ] bX [ ] k 1 2 1 2 − − π Shift j 2 kn / N x n ([ n ]) N e X k [ ] 0 0 − Time Reversal x ([ n ]) N * [ ] X k − N 1 ∑ − Convolution x m h n [ ] ([ m ]) X k H k [ ] [ ] N = m 0 − N 1 1 ∑ − x n w n [ ] [ ] Multiplication X r W [ ] ([ k r ]) N N = r 0 61
Circular Shifting Sequences 62
Digital Filters 63
Digital Filters • digital filter is a discrete-time linear, shift invariant system with input-output relation ∞ ∑ = ∗ = − y n [ ] x n [ ] h n [ ] x m h n [ ] [ m ] =−∞ m = ⋅ Y z ( ) X z ( ) H z ( ) H e ω is the system function ( 系统函数 ) with as the j • H z ( ) ( ) complex frequency response ( 频率响应 ) ω = ω + ω j j j H e ( ) H e ( ) jH e ( ) r i ω ω = ω j j j j arg[ H e ( )] H e ( ) H e ( ) e ω ω ω = + j j j log H e ( ) log H e ( ) j arg[ H e ( )] ω ω = j j log H e ( ) Re[log H e ( )] ω ω = j j arg[ H e ( )] Im[log H e ( )] 64
Digital Filters • causal linear shift-invariant ⇒ h [ n ]=0 for n <0 • stable system ⇒ every bounded input produces a bounded output ⇒ a necessary and sufficient condition for stability and for the H e ω j existence of ( ) ∞ ∑ < ∞ h n [ ] =−∞ n 65
Digital Filters • input and output satisfy linear difference equation ( 线性差分 方程 ) of the form N M ∑ ∑ − − = − y n [ ] a y n [ k ] b x n [ r ] k r = = k 1 r 0 • evaluating z-transforms of both sides gives: N M ∑ ∑ − − = − k r Y z ( ) a z Y z ( ) b z X z ( ) k r = = 1 0 k r M ∑ − r b z r ( ) Y z = = = r 0 H z ( ) N ∑ X z ( ) − − k 1 a z k = k 1 66
Digital Filters • H ( z ) is a rational function of z -1 with M zeros and N poles M ∏ − − 1 A (1 c z ) r = = r 1 H z ( ) N ∏ − − 1 (1 d z ) k = k 1 • converges for | z |> R 1 , with R 1 <1 for stability ⇒ all poles of H ( z ) inside the unit circle for a stable, causal system 67
Ideal Filter Responses 68
FIR System • If a k =0, all k , then M ∑ = − = + − + + − ⇒ y n [ ] b x n [ r ] b x n [ ] b x n [ 1] ... b x n [ M ] r 0 1 M = r 0 = ≤ ≤ 1) h n [ ] b 0 n M n = 0 otherwise M M ∑ ∏ − − = = − 1 2) ⇒ M zeros n H z ( ) b z (1 c z ) n m = = n 0 m 1 = ± − 3) if (symmetric, anti-symmetric) h n [ ] h M [ n ] ω ω − ω = j j j M /2 ( ) ( ) H e A e e ω = j ( ) A e real(symmetric), imaginary(anti-symmetric) 69
Linear Phase Filter • no signal dispersion ( 散布 ) because of non-linear phase ⇒ precise time alignment of events in signal 70
FIR Filters • cost of linear phase filter designs – can theoretically approximate any desired response to any degree of accuracy – requires longer filters than non-linear phase designs • FIR filter design methods – window approximation ⇒ analytical, closed form method – frequency sampling approximation ⇒ optimization method – optimal (minimax error) approximation ⇒ optimization method 71
Matlab FIR Design 72
Lowpass Filter Design Example 73
FIR Implementation • linear phase filters can be implemented with half the multiplications (because of the symmetry of the coefficients) 74
IIR Systems N M ∑ ∑ = − + − [ ] [ ] [ ] y n a y n k b x n r k r = = k 1 r 0 • y [ n ] depends on y [ n -1],…, y [ n - N ] as well as x [ n ], …, x [ n - M ] • for M<N M ∑ − r b z r N A ∑ = = = r 0 k H z ( ) (partial fraction expansion) − − N 1 ∑ 1 d z − − = k k 1 1 a z k k = k 1 N ∑ = n (for casual systems) h n [ ] A d ( ) u n [ ] k k = k 1 an infinite duration impulse response 75
IIR Filters • IIR filter issues – efficient implementations in terms of computations – can approximate any desired magnitude response with arbitrarily small error – non-linear phase ⇒ time dispersion of waveform 76
IIR Design Methods • Analog filter design – Butterworth designs: maximally flat amplitude – Bessel designs: maximally flat group delay – Chebyshev designs: equi-ripple in either passband or stopband – Elliptic designs: equi-ripple in both passband and stopband Transform to digital filter • – Impulse invariant transformation 冲击不变法 • match the analog impulse response by sampling • resulting frequency response is aliased version of analog frequency response – Bilinear transformation 双线性变换法 • use a transformation to map an analog filter to a digital filter by warping the analog frequency scale (0 to infinity) to the digital frequency scale (0 to pi) • use frequency pre-warping to preserve critical frequencies of transformation (i.e., filter cutoff frequencies) 77
Matlab Elliptic Filter Design 78
Matlab Elliptic Filter Design 79
IIR Filter Implementation 80
IIR Filter Implementation 81
IIR Filter Implementation 82
Sampling 83
Sampling of Waveforms = − ∞ < < ∞ x n [ ] x nT ( ), n a = ↔ = = µ F 8000Hz T 1/ 8000 125 sec s = ↔ = = µ F 10000Hz T 1/10000 100 sec s = ↔ = = µ F 16000Hz T 1/16000 62.5 sec s = ↔ = = µ F 20000Hz T 1/ 20000 50 sec s 84
The Sampling Theorem If a signal x a ( t ) has a bandlimited Fourier transform X a ( jΩ ) such that X a ( jΩ ) =0 for Ω ≥ 2πF N , then x a ( t ) can be uniquely reconstructed from equally spaced samples x a ( nT ), - ∞< n <∞, if 1/ T ≥ 2 F N ( F S ≥ 2 F N ) (A-D or C/D converter) 85
Sampling Theorem Equations 86
Sampling Theorem Interpretation 87
Sampling Rates • F N = Nyquist frequency (highest frequency with significant spectral level in signal) • must sample at least twice the Nyquist frequency to prevent aliasing (frequency overlap) – telephone speech (300-3200 Hz) ⇒ F S =6400 Hz – wideband speech (100-7200 Hz) ⇒ F S =14400 Hz – audio signal (50-21000 Hz) ⇒ F S =42000 Hz – AM broadcast (100-7500 Hz) ⇒ F S =15000 Hz • can always sample at rates higher than twice the Nyquist frequency (but that is wasteful of processing) 88
Recovery from Sampled Signal • If 1/ T > 2 F N , the Fourier transform of the sequence of samples is proportional to the Fourier transform of the original signal in the baseband, i.e., π 1 Ω = Ω Ω < j T X e ( ) X ( j ), a T T • can show that the original signal can be recovered from the sampled signal by interpolation using an ideal LPF of bandwidth π /T, i.e., π − ∞ sin( ( ) / ) ∑ t nT T = x t ( ) x nT ( ) π − a a ( t nT ) / T =−∞ n – digital-to-analog converter 89
Decimation( 抽取 ) and Interpolation( 内插 ) of Sampled Waveforms • CD rate (44.06 kHz) to DAT rate (48 kHz)—media conversion • Wideband (16 kHz) to narrowband speech rates (8kHz, 6.67 kHz)—storage • oversampled to correctly sampled rates--coding 90
Decimation and Interpolation of Sampled Waveforms 91
Decimation • Standard Sampling: begin with digitized signal • can achieve perfect recovery of x a ( t ) from digitized sample under these conditions 92
Decimation • to reduce sampling rate of sampled signal by factor of M ≥ 2 • to compute new signal x d [ n ] with sampling rate = = = F ' 1/ T ' 1/ ( MT ) F / M s s such that x d [ n ]= x a ( nT’ ) with no aliasing • one solution is to downsample x [ n ]= x a ( nT ) by retaining one out of every M samples of x [ n ] , giving x d [ n ]= x [ nM ] 93
Decimation • need F s ’ ≥ 2 F N to avoid aliasing for M =2 • when F s ’ < 2 F N , we get aliasing for M =2 94
Decimation • to decimate by factor of M with no aliasing, need to ensure that the highest frequency in x [ n ] is no greater than F s /(2 M ) • thus we need to filter x [ n ] using an ideal lowpass filter with response ω < π 1 / M ω = j H ( e ) d π < ω ≤ π 0 / M • using the appropriate lowpass filter, we can downsample the reuslting lowpass-filtered signal by a factor of M without aliasing 95
Interpolation • assume we have x [ n ]= x a ( nT ) (no aliasing) and we wish to increase the sampling rate by the integer factor of L • we need to compute a new sequence of samples of x a ( t ) with period T ’’= T / L , i.e., x i [ n ]= x a ( nT’’ )= x a ( nT/L ) • It is clear that we can create the signal x i [ n ]= x [ n/L ] for n = 0, ± L , ± 2 L , … but we need to fill in the unknown samples by an interpolation process can readily show that what we want is • π − ∞ sin( ( nT '' kT ) / T ) ∑ = = [ ] ( '') ( ) x n x nT x kT π − i a a ( nT '' kT ) / T =−∞ k • equivalently with T ’’= T / L, x [ n ]= x a ( nT ) , we get π − ∞ sin( ( / n L k )) ∑ = = x n [ ] x nT ( '') x k [ ] π − i a ( / n L k ) =−∞ k which relates x i [ n ] to x [ n ] directly 96
Interpolation implementing the previous equation by filtering the upsampled sequence • = ± ± [ / ] 0, , 2 ,... x n L n L L = x n [ ] u 0 otherwise x u [ n ] has the correct samples for n = 0, ± L , ± 2 L , … , but it has zero-valued • samples in between (from the upsampling operation) The Fourier transform of x u [ n ] is simply • e Ω • Thus is periodic with two periods, namely with period 2π/ L due to j T '' ( ) X u upsampling and 2π due to being a digital signal 97
Interpolation 98
Interpolation • Original signal, x [ n ] , at sampling period, T , is first upsampled to give signal x u [ n ] with sampling period T ’’= T / L • lowpass filter removes images of original spectrum giving x i [ n ]= x a ( nT’’ )= x a ( nT/L ) 99
SR Conversion by Non-Integer Factors • T ’= MT / L ⇒ convert rate by factor of M / L • need to interpolate by L , then decimate by M (why can’t it be done in the reverse order?) – can approximate almost any rate conversion with appropriate values of L and M – for large values of L , or M , or both, can implement in stages, i.e., L =1024, use L 1=32 followed by L 2=32 100
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