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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. | THE HILBERT TRANSFORM 131 N 128 n 0 1. N k 0 1. N x n 0 1 if n 0 N 0 if n 2 T T N -1 if n 0 if n N 0 16 32 48 64 80 96 112 128 n a X k N X x n exPi-j-2-n- -N-k n 0 Imaginary 1 0.5 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 N 0 if k N N j X k if k N c Figure 8-1 Example of the Hilbert transform. 132 DISCRETE-SIGNAL ANALYSIS AND DESIGN Real 1 0.5 Re XH k o -0.5 -1 0 16 32 48 64 80 96 112 128 k d N-1 xh n k 0 XH k -exp e xh n x n f xh1 n 0.25-xh n 1 0.5xh n 0.25xh n 1 xh2 n 025xh1 n 1 0.5xh1 n 0.25xh1 n 1 g xh2 n JL x n 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 defined 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 defined 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 filtering 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 0 which is 0 90 is not always the same thing as the conjugate of 0. If the angle is 30 its conjugate is 30 but its quadrature .
