数字信号处理(英文版)课后习题答案4.docVIP

(Partial) Solutions to Assignment 4 pp.81-82 Discrete Fourier Series (DFS) Discrete Fourier Transform (DFT) , k=0,1,...N-1 , n=0,1,...N-1 Discrete Time Fourier Transform (DTFT) is periodic with period=2π Fourier Series (FS) Fourier Transform (FT) ---------------------------------------------------- 2.1 Consider a sinusoidal signal Q2.1 Consider a sinusoidal signal that is sampled at a frequency =2 kHz a). Determine an expressoin for the sampled sequence , and determine its discrete time Fourier transform b) Determine c) Re-compute from and verify that you obtain the same expression as in (a) a). ans: = where and Using the formular: b) ans: where c). ans: Let be the sample function. The Fourier transform of is Using the relationship or where Consider only the region where ( or therefore where END ----------------------------- 2.3 For each shown, determine where is the sampled sequence. The sampling frequence is given for each case. (b) Hz (d) Hz theory: the relationship between DTFT and FT is where or b. ans: d. ans: omitted (using the same method as above) ---------------------------------------------------- 2.4 In the system shown, let the sequence be and the sampling frequency be kHz. Also let the lowpass filter be ideal, with bandwidth (a). Determine an expression for Also sketch the frequency spectrum (magnitude only) within the frequency range (b) Determine the output signal (a) ans From class notes, we have where is an ZOH interpolation function and We can write Firstly, to find where It can be found as Secondly, find This can be solved either by FT or DTFT. We can write where and Using the formula: we have Using the formula,: we have from DTFT of y[n] Note the above expression is two pulses at and -the scaling factor is: where Therefore, where (b) ans: After the ideal LPF, the Fourier transform of Take inverse Fourier transform of , the output signal is: Note both the and θ terms are introd