数字信号处理(英文版)课后习题问题详解4

实用标准 文档大全 (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) 实用标准

文档大全 Fourier Transform (FT) ---------------------------------------------------- 2.1 Consider a sinusoidal signal Q2.1 Consider a sinusoidal signal

that is sampled at a frequency sF=2 kHz

a). Determine an expressoin for the sampled sequence , and determine its discrete time Fourier transform b) Determine

c) Re-compute ()X from ()XF and verify that you obtain the same expression as in (a)

a). ans:

= where and 实用标准

文档大全 Using the formular:

b) ans:

where

c). ans:

Let be the sample function.

The Fourier transform of is

Using the relationship or 实用标准

文档大全 where Consider only the region where ( or therefore

where END

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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: 实用标准

文档大全 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 实用标准 文档大全 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, 实用标准 文档大全 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 introduced by ZOH function where is introduced because is non-ideal and θ 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 signal

and let the sampling frequency Hz. (a) Determine the continuous time signal after the reconstruction. (a) ans: Assuming (ZOH+ ideal 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. 实用标准 文档大全 ---------------------------------------------------- Q 2.6 In the system shown, determine the output signal for each of the following input signal Assume the sampling 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 equation 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) 实用标准

文档大全 (b) ans: The variance of the output of the filter is given by

Therefore ---------------------------------------------------- ---------------------------------------------------- ----------------------------------------------------

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