By Alan V. Oppenheim
For senior/graduate-level classes in Discrete-Time sign Processing.
Discrete-Time sign Processing, 3rd Edition is the definitive, authoritative textual content on DSP – perfect for people with introductory-level wisdom of signs and structures. Written by means of well-known DSP pioneers, it offers thorough remedy of the basic theorems and houses of discrete-time linear structures, filtering, sampling, and discrete-time Fourier research. by means of targeting the overall and common innovations in discrete-time sign processing, it continues to be important and correct to the hot demanding situations bobbing up within the field.
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Additional resources for Discrete-Time Signal Processing (3rd Edition) (Prentice-Hall Signal Processing Series)
6) k=−∞ Equation (4. 6) is the specified dating among the Fourier transforms of the enter and the output of the impulse educate modulator in determine four. 2(a). Equation (4. 6) states that the Fourier remodel of xs (t) involves periodically repeated copies of X c (j ), the Fourier remodel of xc (t). those copies are shifted through integer multiples of the sampling frequency, after which superimposed to provide the periodic Fourier rework of the impulse teach of samples. determine four. three depicts the frequency-domain illustration of impulse teach sampling. determine four. 3(a) represents a bandlimited Fourier remodel having the valuables that X c (j ) = zero for | | ≥ N . determine four. 3(b) exhibits the periodic impulse teach S(j ), and determine four. 3(c) indicates X s (j ), the results of convolving 1 . it really is glaring that once X c (j ) with S(j ) and scaling by way of 2π s − N ≥ N, or s ≥2 N, (4. 7) Section four. 2 Frequency-Domain illustration of Sampling 157 Xc (j⍀) 1 – ⍀N ⍀N ⍀ (a) S( j ⍀) 2 T –2⍀s – ⍀s ⍀s 2⍀s 3⍀s ⍀ ⍀N ⍀s (⍀s – ⍀N) 2⍀s 3⍀s ⍀ zero (b) Xs (j ⍀) 1 T –2⍀s – ⍀s – ⍀N (c) 1 T Xs (j ⍀) (⍀s – ⍀N) ⍀s 2⍀s ⍀ (d) determine four. three Frequency-domain illustration of sampling within the time area. (a) Spectrum of the unique sign. (b) Fourier rework of the sampling functionality. (c) Fourier rework of the sampled sign with s > 2 N . (d) Fourier rework of the sampled sign with s < 2 N . as in determine four. 3(c), the replicas of X c (j ) don't overlap, and consequently, after they are additional jointly in Eq. (4. 6), there is still (to inside of a scale issue of 1/T ) a reproduction of X c (j ) at every one integer a number of of s . for this reason, xc (t) might be recovered from xs (t) with an excellent lowpass ﬁlter. this can be depicted in determine four. 4(a), which exhibits the impulse educate modulator via an LTI procedure with frequency reaction H r (j ). For X c (j ) as in determine four. 4(b), X s (j ) will be as proven in determine four. 4(c), the place it's assumed that s > 2 N . in view that X r (j ) = H r (j )X s (j ), (4. eight) Chapter four 158 ϱ s (t) = n = –ϱ ϫ xc (t) Sampling of Continuous-Time signs ␦(t – nT ) Hr (j ⍀) xs (t) xr (t) (a) Xc (j ⍀) 1 ⍀N –⍀N ⍀ (b) Xs (j ⍀) 1 T – ⍀s – ⍀N ⍀s > 2⍀N ⍀N ⍀s ⍀ (⍀s – ⍀N) (c) Hr (j ⍀) ⍀N Յ ⍀c Յ (⍀s – ⍀N) T – ⍀c ⍀c ⍀ (d) Xr (j ⍀) 1 – ⍀N ⍀N determine four. four unique restoration of a continuous-time sign from its samples utilizing a fantastic lowpass ﬁlter. ⍀ (e) it follows that if H r (j ) is a perfect lowpass ﬁlter with achieve T and cutoff frequency such that N ≤ c ≤( s − N ), c (4. nine) then X r (j ) = X c (j ), (4. 10) as depicted in determine four. 4(e) and hence xr (t) = xc (t). If the inequality of Eq. (4. 7) doesn't carry, i. e. , if s < 2 N , the copies of X c (j ) overlap, in order that after they are additional jointly, X c (j ) isn't any longer recoverable by way of Section four. 2 Frequency-Domain illustration of Sampling 159 lowpass ﬁltering. this can be illustrated in determine four. 3(d). subsequently, the reconstructed output xr (t) in determine four. 4(a) is expounded to the unique continuous-time enter via a distortion often called aliasing distortion, or, extra easily, aliasing.