DISCRETE-SIGNAL ANALYSIS AND DESIGN- P30

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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P30:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former 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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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 1 x(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 Imaginary Im(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 133 chapter we will continue to use DFT and IDFT and stay focused on the main objective, understanding the Hilbert transform. Why do the samples in Fig. 8-1f and h bunch up at the two ends and in the center to produce the large peaks? The answer can be seen by comparing 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 are the Fourier series constituents of the symmetrical square wave in Fig. 8-1a. In Fig. 8-1d we see a collection of ( − cosine) waves as deÞned in Fig. 2-2b. These ( − cosine) wave harmonic amplitudes accumulate at the endpoints and the center exactly as Fig. 8-1f and h verify. As the harmonics are attenuated, the peaks are softened. The smoothing also tends to equalize adjacent amplitudes slightly. The peaks in Fig. 8-1h rise about 8 dB above the square-wave amplitude, which is almost always too much. There are various ways to deal with this. One factor is that the square-wave input is unusually abrupt at the ends and center. Smoothing (equivalent to lowpass Þltering) of the input signal x (n), is a very useful approach as described in Chapter 4. This method is usually preferred in circuit design. It is useful to keep in mind, especially when working with the HT, that the quadrature of θ, which is θ ± 90◦ , is not always the same thing as the conjugate of θ. If the angle is +30◦ , its conjugate is −30◦ , but its quadrature is +30◦ ± 90◦ = +120◦ or −60◦ . The HT uses θ◦ ± 90◦ . For example, Fig. 8-1b shows +j 0.5 at k = 127. The HT multiplies this by + j to get a real value of − 0.5 in part (d) at k = 127. This is a quadrature positive phase shift at negative frequency k = 127. A similar event occurs at k = 1. Also, at k = 0 and N the phase jump is 180◦ from ± j to ∓j , and the same, although barely noticeable, at N /2. (Use a highly magniÞed vertical scale in Mathcad.) A different example is shown in Fig. 8-2. The baseband signal in part (b) is triangular in shape, and this makes a difference. The abrupt changes in the square wave are gone, the baseband spectrum (d) contains only real cosine-wave harmonics, and the Hilbert transform (f) contains only sine-wave harmonics. The sharp peaks in xh(n) that we see in Fig. 8-1g disappear in Fig. 8-1h. However, for equal peak-to-peak ampli- tude, the square wave of Fig. 8-1 has 4.8 dB more average power than the triangular wave of Fig. 8-2. It is not clear what practical advantage
134 DISCRETE-SIGNAL ANALYSIS AND DESIGN N := 128 n := 0, 1 .. N k := 0, 1 .. N x(n) := 3 if n = 0 3 − 12·n if n ≥ 1 N −9 + 12· n if n > N N 2 3 if n = N (a) 2 0 x(n) −2 −4 0 16 32 48 64 80 96 112 128 n (b) N−1 X(k) := 1 N ∑⋅ n=0 x(n)⋅exp −j⋅2⋅π⋅ n ⋅k N (c) 1.5 1 Re(X(k)) 0.5 0 −0.5 0 16 32 48 64 80 96 112 128 k (d) Figure 8-2 Hilbert transform using a triangular waveform. the triangular wave would have. The importance of peak amplitude limits and peak power limits in circuit design must always be kept in mind. BASIC PRINCIPLES OF THE HILBERT TRANSFORM There are many types of transforms that are useful in electronics work. The DFT and IDFT are well known in this book because they transform back
THE HILBERT TRANSFORM 135 xH(k) := −j ⋅ X(k) if k < N 2 N 0 if k = 2 j ⋅ X(k) if k > N 2 (e) 1 0.5 Im(XH(k)) 0 −0.5 −1 0 16 32 48 64 80 96 112 128 k (f ) N−1 xh(n) := ∑ n=0 (XH(k)⋅exp j⋅2⋅π⋅ k ⋅k N (g) 4 3 2 Re(xh(n)) 1 x(n) 0 −1 −2 −3 −4 0 20 40 60 80 100 120 n (h) Figure 8-2 (continued ) and forth between the discrete time-domain signal x (n) and the discrete frequency-domain spectrum X (k ). Another very popular transform converts a linear differential equation into a linear algebraic equation. For example, consider the differential