2.161 Signal Processing: Continuous and Discrete Fall 2008

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1 MIT OpenCurseWare Signal Prcessing: Cntinuus and Discrete Fall 2008 Fr infrmatin abut citing these materials r ur Terms f Use, visit:

2 Massachusetts Institute f Technlgy Department f Mechanical Engineering Signal Prcessing - Cntinuus and Discrete Fall Term 2008 Lecture 10 1 Reading: Class Handut: Sampling and the Discrete Furier Transfrm Prakis & Manlakis (4th Ed.) Secs , Sec. 7.1 Oppenheim, Schafer & Buck (2nd Ed.) Secs , Secs The Sampling Therem Given a set f samples {f n } and its generating functin f(t), an imprtant questin t ask is whether the sample set uniquely defines the functin that generated it? In ther wrds, given {f n } can we unambiguusly recnstruct f(t)? The answer is clearly n, as shwn belw, where there are bviusly many functins that will generate the given set f samples. B J In fact there are an infinity f candidate functins that will generate the same sample set. The Nyquist sampling therem places restrictins n the candidate functins and, if satisfied, will uniquely define the functin that generated a given set f samples. The therem may be stated in many equivalent ways, we present three f them here t illustrate different aspects f the therem: A functin f(t), sampled at equal intervals ΔT, can nt be unambiguusly recnstructed frm its sample set {f n } unless it is knwn a-priri that f(t) cntains n spectral energy at r abve a frequency f π/δt radians/s. In rder t uniquely represent a functin f(t) by a set f samples, the sampling interval ΔT must be sufficiently small t capture mre than tw samples per cycle f the highest frequency cmpnent present in f(t). There is nly ne functin f(t) that is band-limited t belw π/δt radians/s that is satisfied by a given set f samples {f n }. 1 cpyright c D.Rwell 2008 J 10 1

3 Nte that the sampling rate, F s =1/ΔT, must be greater than twice the highest cyclic frequency F max in f(t). Thus if the frequency cntent f f(t) is limited t Ω max radians/s (r F max cycles/s) the sampling interval ΔT must be chsen s that π ΔT < Ωmax r equivalently 1 ΔT < 2Fmax The minimum sampling rate t satisfy the sampling therem F N =Ω max /π samples/s is knwn as the Nyquist rate. 1.1 Aliasing Cnsider a sinusid f(t) = A sin(at + φ) sampled at intervals ΔT, s that the sample set is {f n } = {A sin(anδt + φ)}, and nting that sin(t) = sin(t +2kπ) fr any integer k, (( ) ) 2πm f n = A sin(anδt + φ) = A sin a + nδt + φ ΔT where m is an integer, giving the fllwing imprtant result: Given a sampling interval f ΔT, sinusidal cmpnents with an angular frequency a and a +2πm/ΔT, fr any integer m, will generate the same sample set. In the figure belw, a sinusid is undersampled and a lwer frequency sinusid, shwn as a dashed line, als satisfies the sample set. f(t) 0 Δ T t

4 This phenmenn is knwn as aliasing. After sampling any spectral cmpnent in F (jω) abve the Nyquist frequency π/δt will masquerade as a lwer frequency cmpnent within the recnstructin bandwidth, thus creating an errneus recnstructed functin. The phenmenn is als knwn as frequency flding since the high frequency cmpnents will be flded dwn int the assumed system bandwidth. One-half f the sampling frequency (i.e. 1/(2ΔT ) cycles/secnd, r π/δt radians/secnd) is knwn as the aliasing frequency, r flding frequency fr these reasns.! F, 6 # F, 6 9 F, 6! F, 6 F, 6 F, 6! F, 6 F, 6 # F, 6! F, 6 The fllwing figure shws the effect f flding in anther way. In (a) a functin f(t) with Furier transfrm F (j Ω) has tw disjint spectral regins. The sampling interval ΔT is chsen s that the flding frequency π/δt falls between the tw regins. The spectrum f the sampled system between the limits π/δt < Ω π/δt is shwn in (b). The frequency cmpnents abve the aliasing frequency have been flded dwn int the regin π/δt < Ω π/δt I = F E C = J, 6 = E = I I F A? J H =? F A J I = E = I I F A? J H =? F A J I 9 F, 6 F, 6 F, 6 F, 6 9 = > 1.2 Anti-Aliasing Filtering: Once a sample set {f n } has been taken, there is nthing that can be dne t eliminate the effects f aliased frequency cmpnents. The nly way t guarantee that the sample set unambiguusly represents the generating functin is t ensure that the sampling therem criteria have been met, either by 10 3

5 1. Selecting a sampling interval ΔT sufficiently small t capture all spectral cmpnents, r 2. Prcessing the cntinuus-time functin f(t) t eliminate all cmpnents at r abve the Nyquist rate. The secnd methd invlves the use f a cntinuus-time prcessr befre sampling f(t). A lw-pass aanti-aliasing filter is used t eliminate (r at least attenuate) spectral cmpnents at r abve the Nyquist frequency. Ideally the anti-aliasing filter wuld have a transfer functin { 1 fr Ω < π/δt H(j Ω) = 0 therwise,. B J. 9 ) J E = E = I E C M F = I I B E J A H F 6 B J = F A H, 6 B? JE K K = EI? HA = E In practice it is nt pssible t design a filter with such characteristics, and a mre realistic gal is t reduce the ffending spectral cmpnents t insignificant levels, while maintaining the fidelity f cmpnents belw the flding frequency. 1.3 Recnstructin f a Functin frm its Sample Set We saw in Lecture 9 that the spectrum F (j Ω) f a sampled functin f (t) is infinite in extent and cnsists f a scaled peridic extensin f F (j Ω) with a perid f 2π/ΔT, i.e. ( ( )) 1 2πn F (j Ω) = F j Ω. ΔT ΔT n= I = F E C = J, 6, 6 F, 6 F, 6 9 " F, 6 F, 6 F, 6 F, 6 F, 6 " F, 6 9 = > If it is assumed that the sampling therem was beyed during sampling, the repetitins in F (j Ω) will nt verlap, and in fact f(t) will be entirely specified by a single perid f F (j Ω). Therefre t recnstruct f(t) we can pass f (t) thrugh an ideal lw-pass filter with transfer functin H(j Ω) that will retain spectral cmpnents in the range π/δt < Ω < π/δt and reject all ther frequencies. 10 4

6 B J 1@ A = HA? I JHK? JE M F = I I BE JA H F, 6 F, 6 B J If the transfer functin f the recnstructin filter is { ΔT fr Ω < π/δt H(j Ω) = 0 therwise, in the absence f aliasing in f (t), that is n verlap between replicatins f F (j Ω) in F (j Ω), the filter utput will be The filter s impulse respnse h(t) is y(t) = F 1 {F (j Ω)H(j Ω)} = F 1 {F (j Ω)} = f(t). h(t) = F 1 {H(j Ω)} = sin (πt/δt ), πt/δt D J ", 6!, 6, 6, 6, 6, 6!, 6 ", 6 J and nte that the impulse respnse h(t) = 0 at times t = ±nδt fr n =1, 2, 3,... (the sampling times). The utput f the recnstructin filter is the cnvlutin f the input functin f (t) with the impulse respnse h(t), f(t) = f (t) h(t) = h(σ) f(t σ)δ(t nδt σ)dσ sin (πσ/δt) n= = f(t σ)δ(t nδt σ)dσ πσ/δt n= sin (π(t nδt )/ΔT ) = f(nδt ), π(t nδt )/ΔT n= 10 5

7 r in the case f a finite data recrd f length N sin (π(t nδt )/ΔT ) f(t) = f n. π(t nδt )/ΔT n=0 This is knwn as the cardinal (r Whittaker) recnstructin functin. It is a superpsitin f shifted sinc functins, with the imprtant prperty that at t = nδt, the recnstructed functin f(t) = f n. This can be seen by letting t = nδt, in which case nly the nth term in the sum is nnzer. Between the sample pints the interplatin is frmed frm the sum f the sinc functins. The recnstructin is demnstrated belw, where a sample set (N = 13) with three nnzer samples is recnstructed. The individual sinc functins are shwn, tgether with the sum (dashed line). Ntice hw the zers f the sinc functins fall at the sample pints The Discrete Furier Transfrm (DFT) We saw in Lecture 8 that the Furier transfrm f the sampled wavefrm f (t) can be written as a scaled peridic extensin f F (j Ω) ( ( )) 1 2nπ F (j Ω) = F j Ω ΔT T n= We nw lk at a different frmulatin f F (j Ω). The Furier transfrm f the sampled functin f (t) F (j Ω) = f (t)e jωt dt = f(t)δ(t nδt )e jωt dt = f(nδt )e jωnδt n= 10 6 n=

8 by reversing the rder f integratin and summatin, and using the sifting prperty f δ(t). We nte: F (j Ω) is a cntinuus functin f Ω, but is cmputed frm the sample pints f(nδt ) in f(t). We have shwn that F (j Ω) is peridic in Ω with perid Ω 0 =2π/ΔT. We nw restrict urselves t a finite, causal wavefrm f(t) in the interval 0 t <nδt,s that it has N samples, and let F (j Ω) = f(nδt )e jωnδt n=0 which is knwn as the Discrete-Time Furier Transfrm (DTFT). As a further restrictin cnsider cmputing a finite set f N samples f F (j Ω) in a single perid, frm Ω = 0 t 2π/ΔT, that is at frequencies 2πm Ω m = fr m =0, 1, 2,...,N 1 NΔT. 9 A G K E I F =? I = F A I E A F A B. 9 F, 6 F, 6 F, 6 and writing F m = F (j Ω m )= F (j 2πm/NΔT ), the DTFT becmes F m = f n e j2πmn/n fr m =0, 1, 2,...,N 1 n=0 where f n = f(nδt ). This equatin is knwn as the Discrete Furier Transfrm (DFT) and relates the sample set {f n } t a set f samples f its spectrum {F m } bth f length N. The DFT can be inverted and the sample set {f n } recvered as fllws: 1 j2πmn/n f n = F m e fr n =0, 1, 2,...,N 1 N m=0 which is knwn as the inverse DFT (IDFT). These tw equatins tgether frm the DFT pair. 10 7

9 The DFT peratins are a transfrm pair between tw sequences {f n } and {F m }. The DFT expressins d nt explicitly invlve the sampling interval ΔT r the sampled frequency interval Ω = 2π/(nΔT ). Simple substitutin int the frmulas will shw that bth F m and f n are peridic with perid N, that is f n+pn = f n and F m+pn = F m fr any integer p. The inverse transfrm is easily demnstrated: ( ) 1 1 j 2πmn/N f n = F m e = fk e e N since = N m=0 m=0 k=0 fk e 1 j 2πm(n k)/n N k=0 m=0 1 = (Nf n ) = f n N { N fr n = k e = m=0 j 2πm(n k)/n 0 therwise. j 2πmk/N j 2πmn/N As in the cntinuus Furier transfrm case, we adpt the ntatins DFT {f n } {F m } {F m } = DFT {f n } {f n } = IDFT {F m } t indicate DFT relatinships. 10 8