tailieunhanh - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P13

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P13: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. | 46 DISCRETE-SIGNAL ANALYSIS AND DESIGN lines are drawn for an e value of 10-3. Again the spectrum of the signal is shown at integer values of k . If we are able to confine our interest to these integer values of k these figures characterize the performance of the DFT for an input that is very close in frequency to an integer k value. Exact values of k give optimum frequency resolution between adjacent values of k which is why they are preferred when possible. It is not always possible as we will see in Chapter 4. Figure 3-1a and b do not tell the entire story. Assume the following x n complex input voltage sequence at frequency k0 in Eq. 3-1 n x n expl j2n k0 I k0 3-1 Using Eq. 1-2 for the DFT N 128 and k 30 to 46 in steps of the phasor frequency response X k is review p. 24 1 A-1 1n X k N - x n exp -j2n k n 0 1 i n n N exp U2nNk0 exp -j2nNkJ n 0 1 A-1 N Sexp j2nNk- k 3-2 n 0 Mathcad finds the real part the imaginary part and the magnitude of the complex exponential phasor at each non integer value of k . Figure 3-1c shows the magnitude in dB on a 0 to -40 dB scale. This is a selectivity curve ratio in dB below the peak for the DFT. At or for example the response is down dB. An input signal at either of these frequencies will show a reduced output at the scalloping effect . At any other input signal frequency k0 that lies between adjacent integer-k values we can repeat Eq. 3-2 to find the spectrum for that k0 and we suggest that the reader experiment with this for additional insight. The last term in Eq. 3-2 is a virtual scalar spectrum analyzer. At each increment of frequency it calculates and then sums for each SPECTRAL LEAKAGE AND ALIASING 47 of 128 time values in Fig. 3-1c the magnitude of X k in dB below the reference level 0 dB . Note that the loop width at 38 is and other loop widths are but the response is zero at 37 and 39. If we calculate the real and imaginary parts of the spectrum we could have a vector .