1、(Partial) Solutions to Assignment 4pp.81-82Discrete Fourier Series (DFS)Discrete Fourier Transform (DFT), k=0,1,.N-1, n=0,1,.N-1Discrete Time Fourier Transform (DTFT)is periodic with period=2Fourier Series (FS)Fourier Transform (FT)-2.1 Consider a sinusoidal signal Q2.1 Consider a sinusoidal signal
2、that is sampled at a frequency =2 kHzsFa). 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 ()X()Fin (a)a). ans: =where and Using the formular: b) ans:where c). ans:
3、Let be the sample function. The Fourier transform of is Using the relationship or where Consider only the region where ( or thereforewhere END-2.3 For each shown, determine where is the sampled sequence. The sampling frequence is given for each case.(b) Hz(d) Hztheory: the relationship between DTFT
4、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
5、 the frequency range (b) Determine the output signal (a) ansFrom class notes, we have where is an ZOH interpolation function and We can writeFirstly, 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 haveUsing the for
6、mula,: we have from DTFT of ynNote 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 terms are introduced by ZOH function where is
7、introduced because is non-ideal and represents the delay of -Q 2.5. We want to digitize and store a signal on a CD, and then reconstruct it at a later time. Let the signaland let the sampling frequency Hz.(a) Determine the continuous time signal after the reconstruction.(a) ans: Assuming (ZOH+ ideal
8、 LPF) is used. This problem can be solved by using the results directly from Q2.4. In Q2.5 there are 3 sinusoidal signals instead of only one in Q2.4. Details of the solutions are omitted.-Q 2.6 In the system shown, determine the output signal for each of the following input signal Assume the sampli
9、ng frequency kHz and the low pass filter (LPF) to be ideal, with bandwidth (b) (d) Ans (b) (d): same as in problem Q2.5. -2.7 Suppose in DAC you want to use a linear interpolation between samples, as shown in the accompanying figure. This reconstructor can be called a first order hold, because the e
10、quation of a line is a polynomial of degree 1(a). Show that with a triangular pulse as shown in the figure(b). Determine an expression for in terms of and (c). In the accompanying figure, let kHz, and the filter be ideal with bandwidth Determine the output Ans: omitted.-2.9 In the following system, let the signal be affected by some random error as shown. The error is white, zero mean, with variance Determine the variance of the error after the filter for each of the filter (b) (b) ans:The variance of the output of the filter is given byTherefore-
