数字信号处理(英文版)课后习题答案4

(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=2πFourier Series (FS)

Fourier Transform (FT)

---------------------------------------------------- 2.1 Consider a sinusoidal signal

Q2.1 Consider a sinusoidal signal

that is sampled at a frequency s F =2 kHz

a). Determine an expressoin for the sampled sequence , and determine its

discrete time Fourier transform

b) Determine

c) Re-compute ()X from ()X F 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

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 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.

----------------------------------------------------

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)

(b) ans:

The variance of the output of the filter is given by

Therefore

----------------------------------------------------

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Less on 1 CABDD BDAAC AB Lesson 2 BCBDC ACAAD BC Lesson 3 CABDA CDABA CD Lesson 4 ACBAB BCDAA BD Less on 5 CABAB DACBB DD Less on 6 CACCC AAADB AA Lesson 7 DCABA BACDA AC Less on 8 BDABD BAABC BC Lesson 9 CDBAA CABAC AD Less on 10 CAABD CBBDC AA Less on 11 AABDD DADDB DB Less on 12 CABAC CDACA AB Lesson 13 ACDAC BDABC AD Less on 14 DBDCC ACCBD BD Less on 15 CADCD DBACA CA Less on 16 ABCCA DDBAB AC Less on 17 BBADA BBDCD CA Less on 18 BABCD CDCCC BA Less on 19 BBCAD AABDD BC Lesson 20 BCADC CCBDB CA Less on 21 BDBBA ADDAB CA Less on 22 CDACB ADBCD AB Less on 23 CADCC DCABC AC

Lesson 25 DBADD CACDB CA Lesson 26 CBCBA CDDAB AC Lesson 27 BCDCC ACCDD DA Less on 28 ADCDA BCADA BD Less on 29 CCADD CCADA BC Lesson 30 CABDD BCCAC DC Less on 31 AABAD BADDC BD Less on 32 BDCBA DBDCA BC Lesson 33 BDBAD BCCDC BA Lesson 34 DCACB DACDB CA Lesson 35 CBCAC ABBDC CD Less on 36 ACBCC ACCDB AC Lesson 37 CABAC DBCDC BD Less on 38 CAABB ACBDD AB Less on 39 BCADA BDDBD BC Less on 40 DCDAC ADDDA DB Less on 41 ACACD CBBBD BC Less on 42 BCCBD BDADC AC Less on 43 DBABC CDDAC BB Lesson 44 AAAAB BBBDC BA Less on 45 CADAC CACDC DC Less on 46 BBDBD ABCDA BD

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