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1、3.5 Band-Limited Discrete-Time SignalsnAn ideal band-limited signal has a spectrum that is zero outside a frequency range 0a|b, that is3 Discrete-Time Signals in the Frequency DomainnA classification of a band-limited discrete-time signal is based on the frequency range where most of the signals ene
2、rgy is concentratednA lowpass discrete-time real signal has a spectrum occupying the frequency range 0|p and has a bandwidth of p3 Discrete-Time Signals in the Frequency DomainnA highpass discrete-time real signal has a spectrum occupying the frequency range 0p| and has a bandwidth of - p nA bandpas
3、s discrete-time real signal has a spectrum occupying the frequency range 0L|H and has a bandwidth of H-L3 Discrete-Time Signals in the Frequency DomainnExample -Consider the sequence xn=(0.5)nnIts DTFT is given below on the left along with its magnitude spectrum shown below on the right3 Discrete-Ti
4、me Signals in the Frequency DomainnIt can be shown that 80% of the energy of this lowpass signal is contained in the frequency range 0|0.5081nHence, we can define the 80% bandwidth to be 0.50813 Discrete-Time Signals in the Frequency Domain3.6 DTFT Computation Using MATLABnThe function freqz can be
5、used to compute the values of the DTFT of a sequence, described as a rational function in the form ofat a prescribed set of discrete frequency points = l3 Discrete-Time Signals in the Frequency DomainnFor example, the statement H = freqz(num, den, w)returns the frequency response values as a vector
6、H of a DTFT defined in terms of the vectors num and den containing the coefficients pi and di, respectively at a prescribed set of frequencies between 0 and 2 given by the vector w 3 Discrete-Time Signals in the Frequency DomainnThere are several other forms of the function freqznThe Program 3_1.m i
7、n the text can be used to compute the values of the DTFT of a real sequencenIt computes the real and imaginary parts, and the magnitude and phase of the DTFT3 Discrete-Time Signals in the Frequency DomainnExample 3.17 Plots of the real and imaginary parts, and the magnitude and phase of the DTFT are
8、 shown on the next slide3 Discrete-Time Signals in the Frequency Domainwrapped phase3 Discrete-Time Signals in the Frequency DomainnIn numerical computation, when the computed phase function is outside the range -, , the phase is computed modulo 2, to bring the computed value to this rangenThus, the
9、 phase functions of some sequences exhibit discontinuities of 2 radians in the plot.3.7 The unwrapped phase functionnFor example3.17, there is a discontinuity of 2 at =0.72 in the phase function.3 Discrete-Time Signals in the Frequency DomainnIn such cases, often an alternate type of phase function
10、that is continuous function of is derived from the original phase function by removing the discontinuities of 2nProcess of discontinuity removal is called unwrapping the phasenThe unwrapped phase function will be denoted as c(),which indicates a continuous function of .3 Discrete-Time Signals in the
11、 Frequency DomainnNote: This discontinuity can be removed using the function unwrap in MATLAB.Wrapped Phase Unwrapped Phase3 Discrete-Time Signals in the Frequency DomainnThe conditions under which the phase function will be continuous function of .X(ej ) = | X(ej ) |ej ( ) (3.19)where ( ) = argX(ej ) (3.20)3 Discrete-Time Signals in the Frequency Domain(3.55)IFExists, ( ) can be defined uneqivocally byWith constraint3 Discrete-Time Signals in the Frequency Domain(3.59)(3.60)will be continuous function of Unwrapped phaseMoreover, if will be odd function of
