FT Representation of DT Signals: Relating FT to DTFT

a) DTFT of x[n]: b) FT of CT signal

Sampling. The figure shown 2 slides earlier: Continuous-time representation of discrete-time signal x[n]

The FT of a sampled signal for different sampling frequencies The FT of a sampled signal for different sampling frequencies. Spectrum of continuous-time signal. Spectrum of sampled signal when s = 3W. Spectrum of sampled signal when s = 2W. (d) Spectrum of sampled signal when s = 1.5W.

Observations: FT of a sampled signal: x(jw) shifted by integer multiples of ws 2)

DTFT of sampled signal x[n] and FT of xd(t) Example 4.9, p366:

The effect of sampling a sinusoid at different rates (Example 4. 9) The effect of sampling a sinusoid at different rates (Example 4.9). (a) Original signal and FT. (b) Original signal, impulse sampled representation and FT for Ts = ¼. (c) Original signal, impulse sampled representation and FT for Ts = 1. (d) Original signal, impulse sampled representation and FT for Ts = 3/2. A cosine of frequency /3 is shown as the dashed line.

E Problem 4.10, p368: Draw the FT of a sampled version of the CT signal having the FT depicted By the following figure for (a) Ts=1/2 and (b) Ts=2. (a) Ts=1/2, ws=4p. (b) Ts=2, ws=p.

Downsampling: Let E

Figure 4. 29 (p. 372) Effect of subsampling on the DTFT Figure 4.29 (p. 372) Effect of subsampling on the DTFT. (a) Original signal spectrum. (b) m = 0 term, Xq(ej), in Eq. (4.27) (c) m = 1 term in Eq. (4.27). (d m = q – 1 term in Eq. (4.27). (e) Y(ej), assuming that W < /q. (f) Y(ej), assuming that W > /q.

Sampling theorem E Example 4.12, p347:

Ideal reconstruction: Spectrum of original signal. Spectrum of sampled signal. (c) Frequency response of reconstruction filter.

Figure 4.36 (p. 377) Ideal reconstruction in the time domain.

Figure 4.37 (p. 377) Reconstruction via a zero-order hold. Ideal reconstruction is not realizable Practical systems could use a zero-order hold block This distorts signal spectrum, and compensation is needed Figure 4.37 (p. 377) Reconstruction via a zero-order hold.

Figure 4.38 (p. 378) Rectangular pulse used to analyze zero-order hold reconstruction.

Figure 4.39 (p. 379) Effect of the zero-order hold in the frequency domain. (a) Spectrum of original continuous-time signal. (b) FT of sampled signal. (c) Magnitude and phase of Ho(j). (d) Magnitude spectrum of signal reconstructed using zero-order hold.

Figure 4.40 (p. 380) Frequency response of a compensation filter used to eliminate some of the distortion introduced by the zero-order hold. Anti-imaging filter.

Figure 4.41 (p. 380) Block diagram of a practical reconstruction system.

Figure 4.43 (p.383) Block diagram for discrete-time processing of continuous-time signals. (a) A basic system. (b) Equivalent continuous-time system.

Idea: find the CT system 0th-order S/H:

If no aliasing, the anti-imaging filter Hc(jw) eliminates frequency components above ws/2, leaving only k=0 terms If anti-aliasing and anti-imaging filters are chosen to compensate the effects of sampling and reconstruction, then

Oversampling: Sampling rate must be greater than Nyquist rate to relax anti-aliasing filter design Let Ws be cutoff frequency of anti-aliasing filter Ha(jw) and W be the maximum frequency of desired signal Then, to avoid aliasing, Due to DSP, noise aliases are not of concern, thus (see figure next slide)

Figure 4.44 (p. 385) Effect of oversampling on anti-aliasing filter specifications. (a) Spectrum of original signal. (b) Anti-aliasing filter frequency response magnitude. (c) Spectrum of signal at the anti-aliasing filter output. (d) Spectrum of the anti-aliasing filter output after sampling. The graph depicts the case of s > 2Ws.

Decimation (downsampling): To relax design of anti-aliasing filter and anti-imaging filters, we wish to use high sampling rates High-sampling rates lead to expensive digital processor Wish to have: High rate for sampling/reconstruction Low rate for discrete-time processing This can be achieved using downsampling/upsampling

Figure 4. 45 (p. 387) Effect of changing the sampling rate Figure 4.45 (p. 387) Effect of changing the sampling rate. (a) Underlying continuous-time signal FT. (b) DTFT of sampled data at sampling interval Ts1. (c) DTFT of sampled data at sampling interval Ts2.

Figure 4.46 (p. 387) The spectrum that results from subsampling the DTFT X2(ej) depicted in Fig. 4.45 by a factor of q. Figure 4.48 (p. 389) Symbol for decimation by a factor of q (downsampling).

Figure 4. 47 (p. 388) Frequency-domain interpretation of decimation Figure 4.47 (p. 388) Frequency-domain interpretation of decimation. (a) Block diagram of decimation system. (b) Spectrum of oversampled input signal. Noise is depicted as the shaded portions of the spectrum. (c) Filter frequency response. (d) Spectrum of filter output. (e) Spectrum after subsampling.

Upsampling (zero padding):

Figure 4. 49 (p. 390) Frequency-domain interpretation of interpolation Figure 4.49 (p. 390) Frequency-domain interpretation of interpolation. (a) Spectrum of original sequence. (b) Spectrum after inserting q – 1 zeros in between every value of the original sequence. (c) Frequency response of a filter for removing undesired replicates located at  2/q,  4/q, …,  (q – 1)2/q. (d) Spectrum of interpolated sequence.

Figure 4. 50 (p. 390) (a) Block diagram of an interpolation system Figure 4.50 (p. 390) (a) Block diagram of an interpolation system. (b) Symbol denoting interpolation by a factor of q.

Figure 4.51 (p. 391) Block diagram of a system for discrete-time processing of continuous-time signals including decimation and interpolation.

FS representation of finite-duration nonperiodic signals Discrete-time periodic signals: DTFS representation Continuous-time periodic signals: FS representation For numerical computation, it is better to have BOTH discrete in time and discrete in frequency

Figure 4.52 (p. 392) The DTFS of a finite-duration nonperiodic signal.

Figure 4. 53 (p. 394) The DTFT and length-N DTFS of a 32-point cosine Figure 4.53 (p. 394) The DTFT and length-N DTFS of a 32-point cosine. The dashed line denotes |X(ej)|, while the stems represent N|X[k]|. (a) N = 32, (b) N = 60, (c) N = 120.

Figure 4.54 (p. 396) Block diagram depicting the sequence of operations involved in approximating the FT with the DTFS.

Figure 4.55 (p. 397) Effect of aliasing.

Figure 4.56 (p. 398) Magnitude response of M-point window.

Figure 4.57 (p. 400) The DTFS approximation to the FT of x(t) = e-1/10 u(t)(cos(10t) + cos(12t). The solid line is the FT |X(j)|, and the stems denote the DTFS approximation NTs|Y[k]|. Both |X(j) and NTs|Y[k]| have even symmetry, so only 0 <  < 20 is displayed. (a) M = 100, N = 4000. (b) M = 500, N = 4000. (c) M = 2500, N = 4000. (d) M = 2500, N = 16,0000 for 9 <  < 13.

Figure 4.58 (p. 404) The DTFS approximation to the FT of x(t) = cos(2(0.4)t) + cos(2(0.45)t). The stems denote |Y[k]|, while the solid lines denote (1/M|Y (j)|. The frequency axis is displayed in units of Hz for convenience, and only positive frequencies are illustrated. (a) M = 40. (b) M = 2000. Only the stems with nonzero amplitude are depicted. (c) Behavior in the vicinity of the sinusoidal frequencies for M = 2000. (d) Behavior in the vicinity of the sinusoidal frequencies for M = 2010.

Figure 4.59 (p. 406) Block diagrams depicting the decomposition of an inverse DTFS as a combination of lower order inverse DTFS’s. (a) Eight-point inverse DTFS represented in terms of two four-point inverse DTFS’s. (b) four-point inverse DTFS represented in terms of two-point inverse DTFS’s. (c) Two-point inverse DTFS.

Figure 4.60 (p. 407) Diagram of the FFT algorithm for computing x[n] from X[k] for N = 8.