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1、CHAPTER 5 THE DISCRETE-TIME FOURIER TRANSFORM5/0 INTRODUCTION/Discrete-time Fourier transform and inverse Fourier transform/ Application of DTFT in discrete-time LTI systems/ Similarities and differences between continuous-time and discrete-time Fourier transforms//5/1 REPRESENTATION OF NONPERIODIC

2、SIGNALS/ THE DISCRETE-TIME FOURIER TRANSFORM/As /Defining a function//Thus/Consequently/As /Differences between the continuous-time and discrete-time Fourier transform are/ periodicity of the discrete-time transform and the finite interval of integration in the synthesis equation//In discrete time/

3、Low frequencies are the values of near even multiple of / high frequencies are those values of near odd multiples of //Example 5/1 Consider the signal/1 0/0 -1/Example 5/2 Consider the signal/for 0 1/Example 5/3 Consider the rectangular pulse/Convergence Issues of the Discrete-Time Fourier Transform

4、/If xn is an infinite duration signal/ we must consider the question of convergence of the infinite summation in the analysis equation//The analysis equation will converge if xn is absolutely summable/ that is/In contrast to the situation for the analysis equation/ there are generally no convergence

5、 issues associated with the synthesis equation//Consider the unit sample xn = n/Approximating n by an integral of complex exponentials with frequencies taken over the interval | W / i/e//This is plotted in the following Figure for several values of W//Approximation to the unit sample using complex e

6、xponentials with frequencies | W/ (a) W = /4/ (b) W = 3/8/ (c) W = /2/ (d) W = 3/4/ (e) W = 7/8/ (f) W = / Note that for W = ///5/2 THE FOURIER TRANSFORM FOR PERIODIC SIGNALS/First consider the Fourier transform of the sequence/To check the validity of this expression/Now consider an arbitrary perio

7、dic sequence xn with period N and with the Fourier series representation //Applying the Fourier transform to both sides/ we obtain/Thus/ the Fourier transform of a periodic signal can be directly constructed from its Fourier coefficients//Example 5/4 Consider the periodic signal/That is/Example 5/5

8、Consider the periodic sample train/Then we can represent the Fourier transform of the signal as/Choosing the interval of summation as 0 n N-1/5/3 PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM/5/3/1 Periodicity of the Discrete-Time Fourier Transform/5/3/2 Linearity of the Fourier Transform/then/5

9、/3/3 Time Shifting and Frequency Shifting/If/then/and/5/3/4 Conjugation and Conjugate Symmetry/If/then/if xn is real valued/5/3/5 Differencing and Accumulation/First-difference//Accumulation//5/3/6 Time Reversal/5/3/7 Time Expansion/If/then/5/3/8 Differentiation in Frequency/5/3/9 Parsevals Relation

10、/Example 5/7 Consider the unit step sequence un//Since/and/Thus/Example 5/8 Consider the sequence xn which is illustrated in the following figure//5/4 THE CONVOLUTION PROPERTY/If/then/The convolution property represents that the Fourier transform of the response of an LTI system to a nonperiodic inp

11、ut are simply the Fourier transform of the input multiplied by the systems frequency response evaluated at the corresponding frequencies//The convolution property maps the convolution operation of two time signals to the multiplication operation of their Fourier transforms//The frequency response ca

12、ptures the change in complex amplitude of the Fourier transform of the input at each frequency //Example 5/8 Consider an LTI system with sample response/The frequency response is/Thus/ for any input xn/ the Fourier transform of the output is/Consequently/Example 5/9 Consider an LTI system with sampl

13、e response/The input to this system is/The output yn = ?/If/Thus/If/Example 5/10 Consider the system/What is the frequency response of the overall system? Where is an ideal low-pass filter with cutoff frequency /4 and unity gain in the passband//The key step//Thus/Since/Consequently/From the convolu

14、tion property/ the overall system has the frequency response//Stopband//5/5 THE MULTIPLICATION PROPERTY/Consider the Fourier transform of yn = x1n x2n/ where the Fourier transforms of x1n and x2n are known//since/Example 5/11 Find the Fourier transform of a signal xn which is the product of x1n and

15、x2n/ where/From the multiplication property/5/6 DUALITY/SUMMARY OF FOURIER SERIES AND TRANSFORM EXPRESSIONS/For the discrete-time Fourier series/ duality between the sequence xn in the time-domain and its Fourier series coefficient f k is//For the continuous-time Fourier transform/ duality between t

16、he signal x(t) in the time-domain and its Fourier transform X(j) is//Duality implies that every property has a dual//There is also a duality between the discrete-time Fourier transform and the continuous-time Fourier series//Example 5/12 Determine the discrete-time Fourier transform of the sequence/

17、Since the Fourier transform of xn is periodic with period 2/ and with the form of square wave/ so we consider signal g(t)/ which is a periodic square wave with period 2/ and with/the Fourier series coefficients of g(t) are (See Table 4/2)/Let T1 = /2/ then we have ak = xk/ taking ak and g(t) into th

18、e analysis equation//Renaming k as n and t as / we have/Replacing n by n / we obtain/Thus/5/7 SYSTEMS CHARACTERIZED BY LINEAR CONSTANT-COEFFICIENT DIFFERENCE EQUATIONS/is a ratio of polynomials in the variable //coefficients of the numerator polynomial = coefficients appearing on the right side of t

19、he difference equation//coefficients of the denominator polynomial = coefficients appearing on the left side of the difference equation//Example 5/13 Consider a causal LTI system that is characterized by the difference equations/and let the input to this system be/Determine the output yn//The form o

20、f the partial-fraction expansion in this case is/So that/Consequently/5/8 DISCRETE-TIME FREQUENCY-SELECTIVE FILTER/5/9 SAMPLING OF DISCRETE-TIME SIGNALS/Impulse-train sampling/spectrum of sampled signal with s 2M/In discrete-time case/ the result of sampling theorem also exist//Example 5/14 Consider

21、 a sequence xn whose Fourier transform/Determine the lowest rate at which xn may be sampled without aliasing//Since/the corresponding sampling frequency is 2/4 = /2//So that/Thus/From the sampling theorem/ we know/5/10 SUMMARY/The Fourier transform for periodic discrete-time signals//The Fourier transform for aperiodic discrete-time signals//Convenient way to obtain the frequency re

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