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
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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:
d. ans:
omitted (using the same method as above)
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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
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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.
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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.
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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.
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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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