DISCRETE-SIGNAL ANALYSIS AND DESIGN- P30

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P30:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P30 THE HILBERT TRANSFORM 131 N := 128 n := 0, 1.. N x(n) := 0 k := 0, 1.. N 1 if n > 0 0 if n = N 2 N −1 if n > 2 0 if n = N 1x(n) 0 −1 0 16 32 48 64 80 96 112 128 n (a) N−1 X(k) := 1 ⋅ ∑ N n=0 x(n)⋅exp −j⋅2⋅π⋅ n ⋅k N 1 0.5 ImaginaryIm(X(k)) 0 −0.5 −1 0 16 32 48 64 80 96 112 128 k (b) XH(k) := −j⋅X(k) if k < N 2 N 0 if k = 2 N j⋅X(k) if k > 2 (c) Figure 8-1 Example of the Hilbert transform.132 DISCRETE-SIGNAL ANALYSIS AND DESIGN 1 0.5 Real Re(XH(k)) 0 −0.5 −1 0 16 32 48 64 80 96 112 128 k (d ) N−1 xh(n) := ∑ k=0 (XH(k))⋅exp j⋅2⋅π⋅ k ⋅n N (e) 4 3 2 (xh(n)) 1 x(n) 0 −1 −2 −3 −4 0 20 40 60 80 100 120 n (f ) xh1(n) := 0.25⋅xh(n − 1) + 0.5⋅xh(n) + 0.25⋅xh(n + 1) xh2(n) := 0.25⋅xh1(n − 1) + 0.5⋅xh1(n) + 0.25⋅xh1(n + 1) (g) 4 3 2 xh2(n) 1 0 x(n) −1 −2 −3 −4 0 20 40 60 80 100 120 n (h) Figure 8-1 (continued ) THE HILBERT TRANSFORM 133chapter we will continue to use DFT and IDFT and stay focused on themain objective, understanding the Hilbert transform. Why do the samples in Fig. 8-1f and h bunch up at the two ends andin the center to produce the large peaks? The answer can be seen bycomparing Fig. 8-1b and d. In Fig. 8-1b we see a collection of (sine)wave harmonics as deÞned in Fig. 2-2c. These sine wave harmonics arethe Fourier series constituents of the symmetrical square wave in Fig.8-1a. In Fig. 8-1d we see a collection of ( − cosine) waves as deÞnedin Fig. 2-2b. These ( − cosine) wave harmonic amplitudes accumulate atthe endpoints and the center exactly as Fig. 8-1f and h verify. As theharmonics are attenuated, the peaks are softened. The smoothing alsotends to equalize adjacent amplitudes slightly. The peaks in Fig. 8-1h riseabout 8 dB above the square-wave amplitude, which is almost always toomuch. There are various ways to deal with this. One factor is that thesquare-wave input is unusually abrupt at the ends and center. Smoothing(equivalent to lowpass Þltering) of the input signal x (n), is a very usefulapproach as described in ...