homework.txt

Guilde To DSP 讲DSP(数字信号处理)原理的好书, 练习英语阅读能力的好机会. ~_^ seabird Nov 13,1

h. RP =  0, IP = -1



CHAPTER 9: APPLICATIONS OF THE DFT

1. There are three signals involved in linear systems: the input signal
(containing information we want to understand), the impulse response
(controlling how the information is modified), and the output signal (a
result of the other two signals).  It is not a coincidence that the DFT has
three main uses.  Match each DFT techniques with its corresponding signal.
Explain how each DFT technique provides a way of understood or dealing with
the associated signal.


2. A scientist acquires 65,536 samples from an experiment at a sampling rate
of 1 MHz.  He knows that the signal contains a sinusoid at 100 kHz.  He
needs to determine is if there is a second sinusoid at 103 kHz.  As a start,
he takes a 65,536 point DFT of the signal.  To his surprise, all he can see
in the spectrum is noise.  He estimates that the signal he is looking for is
4 times lower in amplitude than the random noise (i.e., SNR = 0.25). He also
estimates that the SNR will need to be at least 3.0 for the signal to be
detected, if present.  To improve the SNR, he decides to break the signal
into segments, and average their spectra.  Arrange your answers to the
following questions in a table. 

a. If he uses a segment length of 16,384 samples, what will be the frequency
resolution (i.e., the spacing between data points) in the averaged spectrum?
Give your answer in hertz.  How many segments will be averaged?  What is the
SNR of the averaged spectrum?  Does this have the required frequency
resolution?  Does this have the required SNR?

b. Repeat for segment lengths of: 8192, 4096, 2048, 1024, 512, 256, and 128.


3. If the scientist in the last problem wanted to improve this experiment,
what advice could you give?  


4. A filter kernel (impulse response) consists of 250 samples, and is
designed to pass all frequencies below 0.11, and block all frequencies above
0.12. To evaluate how this filter performs, you want to closely inspect its
frequency response over this range.  To do this, you pad the impulse
response with zeros to make the total length 256 samples, and then take the
DFT. 

a. How may data points are spread over the range of interest? 
b. Repeat using a DFT length of 2048. 
c. Repeat using a DFT length of 2 to the 50th power.  
d. Is there anything limiting how many samples can be placed over this range
of interest? How does this relate to the DTFT? Explain.


5. A signal containing 1000 points is to be convolved with a signal
containing 128 points.

a. What is the length of the resulting signal?
b. If frequency domain convolution is used, what length of DFT is
appropriate?
c. If a 1024 point DFT is used, how many samples are correct, and how many
are corrupted by circular convolution?
d. Repeat (a) to (c) for the two signals having 490 samples and 23 samples. 



CHAPTER 10: PROPERTIES OF THE DFT

1. If x[n] has the frequency domain: Xreal[f] and Ximag[f], and y[n] has the
frequency domain: Yreal[f] and Yimag[f], calculate the frequency domain of
the following signals:

a. x[n]/5
b. 5.5y[n]
c. x[n] + y[n] 
d. 3.14x[n] + y[n]/3.14
e. ax[n] + by[n], where a and b are constants


2. If x[n] has the frequency domain: Xmag[f] and Xphase[f], and y[n] has the
frequency domain: Ymag[f] and Yphase[f], calculate the frequency domain of
the following signals:

a. x[n-2]
b. 1.2x[n-1]
c. y[n+2]/10
d. ax[n-b],  where a and b are constants


3. Regarding the signals in problems 1 and 2:

a. In problem 1, why would it be difficult to calculate the frequency domain
of: x[n-2]?  
b. In problem 2, why would it be difficult to calculate the frequency domain
of x[n] + y[n]?
c Complete the following statements describing this situation: 
"When time domain signals are added, the frequency domain is easiest to
understand when in [fill in the blank] form."
"When time domain signals are shifted, the frequency domain is easiest to
understand when in [fill in the blank] form."
"When time domain signals are scaled, the frequency domain is easiest to
understand when in [fill in the blank] form."
 

4. Suppose that the frequency spectrum in Fig 9-2 represents a digital
signal with a sampling rate of 160 Hz. Sketch the digital frequency spectrum
for frequencies between -2.0 and 2.0.  


5. A digital low-pass filter passes all frequencies below 0.1, and blocks
all frequencies above. 

a. Sketch this frequency response, showing all frequencies between -1.5 and
1.5.
b. If the filter kernel is multiplied by a sine wave with a frequency of
0.3, sketch the new frequency response.
c. Based on the method in (b), describe an algorithm for converting a low-
pass filter with cutoff frequency, fc, into a bandpass filter with a center
frequency, fcenter, and a bandwidth, BW.
d. If the low-pass filter kernel is multiplied by a cosine wave with a
frequency of 0.5, sketch the new frequency response.
e. Based on the method in (d), describe an algorithm for converting a low-
pass filter with cutoff frequency, flp, into a high-pass filter with a
cutoff frequency, fhp. 
f. Why must a cosine wave be used in (e) instead of a sine wave?


6. To represent the sound of a human voice, a signal only needs to contain
frequencies between 100 Hz and 4 kHz.  In comparison, high-fidelity music
must contain all the frequencies that humans can hear, i.e., 20 Hz to 20
kHz.  

a. Sketch and label the frequency spectra of these two signals, including
the negative frequencies. Assume all the frequencies have the same
amplitude.
b. A DC bias is often added to audio signals in electronic circuits.  This
is to make the voltage representing the signal always have a positive value
(see Fig. 10-14a). Repeat (a) assuming that each audio signal has such a DC
bias.
c. Sketch the frequency spectrum of a voice signal multiplied by a sine wave
at 1 MHz.  Repeat for a voice signal plus DC bias.
d. Sketch the frequency spectrum of a music signal multiplied by a sine wave
at 1.1 MHz.  Repeat for a music signal plus DC bias.
e. If the signal in (c) is transmitted from a radio station, what band of
frequencies must be assigned by the government to avoid interference with
other radio applications?  Repeat for the signal in (d). 
f. If a frequency band of 50.1 to 50.2 MHz were available, how many voice
signals could be simultaneously transmitted?  How many high-fidelity music
signals?  Explain how the signals would be prepared for this simultaneous
transmission. 


7. The "no bias" signal in (c) of the last problem is filtered to remove all
frequencies below 1 MHz. The result is called "single-sideband" (SSB)
modulation. 

a. Sketch the single-sideband frequency spectrum. 
b. Does this contain the same information as the original voice signal?  
c. A disadvantage of SSB is that it requires more complicated modulation and
demodulation electronics. What would be the advantage of using single-
sideband modulation over AM?  
d. If the lower sideband were retained instead, would the information in the
original signal still be preserved? Explain. 



CHAPTER 11: FOURIER TRANSFORM PAIRS

1. Give the mathematical equation for the frequency domain corresponding to
the following waveforms.  You can state your answer in either rectangular or
polar form. Example: x[n] = delta[n], answer: Mag X[f] = 1, Phase X[f] = 0.

a. x[n] = 2delta[n-2]
b. x[n] = sin(2 pi n 14 / 256)
c. x[n] = cos(2 pi n 0.2)
d. x[n] = 1 for 10 < n < 18, 0 otherwise
e. The signal in (d) convolved with itself
f. A Gaussian centered at sample 100, with a standard deviation of 20.
g. The signal in (f) multiplied by a cosine wave of frequency 0.3.


2. Those experienced in DSP and electronics can approximate the highest
frequency contained in a signal simply by look at its graph.  This problem
will help you master this useful skill.  In general, the most rapidly
changing sections of a signal will correspond to the highest frequency
contained in the signal.  As an example, the sinc function in Fig. 11-5a has
a period of about 11 samples.  This corresponds to the highest frequency
present in the signal being about 1/11 = 0.09.  In a more realistic example,
Figs. (c) and (d) show waveforms that resemble one-half cycle of a sinusoid
being completed over 16 cycles.  This corresponds to one cycle every 32
samples, or a highest frequency of 1/32 = 0.03. Use this method to estimate
the highest frequency component in the following signals:
  
a. Fig. 7-14, y[n]
b. Fig. 8-8b
c. Fig. 9-7a
d. Fig. 11-6b-d
e. Fig. 3-5a (for "time" in seconds)
f. Fig. 10-14a
g. Fig. 13-8a
h. A signal where each sample is obtained from a random number generator.
i. In some of the above figures, the actual frequency spectra are also
given. Using this information, approximately how accurate is this method?
(Choose from 1%,3%,10%, or 30%)


3. A signal contains 16 consecutive samples with a value of 1, with the
remainder of the samples having a value of zero.  

a. What are the frequencies of the first four zeros crossings in the
frequency domain?  
b. What are the periods of these frequencies?
c. Make a sketch showing how sinusoids at the first two zero crossings fit
evenly inside the rectangular pulse.

 
4.  A discrete sinusoid of frequency 0.125 is half-wave rectified (that is,
all the negative valued samples are set to zero). 

a. Sketch the original time domain signal and its frequency spectrum.
b. Sketch the rectified time domain signal and its frequency spectrum (don't
worry about the exact amplitude of the harmonics).
c. Which harmonics of the rectified signal will be aliased?
d. At what frequency does the 5th harmonic appear?
e. At what frequency does the 10th harmonic appear?
f. At what frequency does the 100th harmonic appear?  


5. Referring to the chirp system in Fig. 11-10:  

a. What is the frequency domain magnitude of the signal in (a)?  Of the
signal in (b)?
b. What is the frequency domain phase of the signal in (a)?  Of the signal
in (b)? (express these answers as equations using the constants alpha and
beta).
c. Using Parseval's relation, what is the relationship between the total
energy contained in the waveforms in (a) and (b)? Explain.
d. Recall from physics that power is equal to the total energy released
divided by the time over which it is released. If the waveform in (a) is one
sample long, and the waveform in (b) is 80 samples long, what power
reduction is provided by this chirp system?

 
6. Figures 11-5 (a) and (e) illustrate a sinc function and a Gaussian,
respectively. Both these functions decrease in amplitude toward the left and
right. However, neither of these functions ever reach a constant value of
zero.

a. Write an equation describing how the amplitude of the sinc's oscillations 
decrease, moving away from the center of symmetry. 
b. Repeat (a) for the Gaussian (estimating the standard deviation from the
figure). 
c. When the main lobes are about the same width [as illustrated in Figs. (a)
and (e)], which of these two functions drops toward zero faster?  To answer
this, evaluate the amplitude of the two functions at 3, 10, 30, 100, and 300
samples from the center of symmetry.
d. For each function, how many samples must you move away from the center of
symmetry before the amplitude fall below the single precision round-off
noise? 
e. Would a Gaussian or a rectangular pulse in the frequency domain be more
likely to result in time domain aliasing? Explain.
f. Would a Gaussian or a rectangular pulse in the time domain be more likely
to result in frequency domain aliasing? Explain.



CHAPTER 12: THE FAST FOURIER TRANSFORM

1. The following spectrum was produced by the real DFT. Generate the
frequency spectrum of the corresponding complex DFT. 

samples 0 to 8 of the real part:        1, 2,-1,-2, 0, 1, 2, 3, 2 
samples 0 to 8 of the imaginary part:   0, 2, 4, 5,-3,-2, 1, 1, 0


2. A time domain signal, consisting of a real part, rex[n], and an imaginary
part, imx[n], has the following complex spectrum:

samples 0 to 7 of the real part:        1, 2,-1,-2, 1, 0, 2, 3
samples 0 to 7 of the imaginary part:   3, 2, 4, 5,-1,-2, 1, 1

a. Separate this spectrum (both the real and imaginary parts) into even and
odd parts. 
b. What would be the spectrum if the values in imx[n] were set to zero?
e. What would be the spectrum if the values in rex[n] were set to zero?
     

3.  Suppose you want to conduct a spectral analysis of a signal containing
1,003,520 samples, and the computer you are using has values of: Kdft = 1
microsecond, and Kfft = 1.5 microsecond. 

a. Calculate the execution time if the signal is broken into 64 sample
segments, and the DFT by correlation algorithm is used.  Repeat for segment
lengths of: 256, 1024, and 4096. (Ignore the calculation time for averaging
the frequency spectra).
b. Repeat (a) using the FFT algorithm.  



CHAPTER 13: CONTINUOUS SIGNAL PROCESSING

1. How short must a pulse be to act as an impulse to the following systems?
(Give an "order-of- magnitude" approximation and justify your reasoning).

a. A high-fidelity music system, designed to reproduce sounds between 20 Hz
and 20 kHz.  
b. An automotive suspension (think about how fast it recovers after the
vehicle hits a bump).
c. An 8 pole Bessel low-pass filter with a cutoff frequency of 1 kHz. (hint:
see Fig. 3-13)
d. The disruption of a galaxy when struck by another galaxy (see the picture
on the cover of the book, and the caption opposite the title page). Hint:
The "time constant" of disruption is the time it takes the expanding gas to
equal the diameter of the galaxy. 


2. Calculate and sketch the convolution of a(t) and b(t). Also provide
sketches showing how each region in the output signal is calculated.  

a. a(t) = 1 for 0 < t < 2, and 0 otherwise 
   b(t) = 1 for 0 < t < 2, and 0 otherwise

b. a(t) = 2 for 0 < t < 1, and 0 otherwise 
   b(t) = 1 for 2 < t < 5, and 0 otherwise

c. a(t) = 1 for 0 < t < 1, and 0 otherwise 
   b(t) = t for 0 < t < 3, and otherwise

d. a(t) = exp(-kt) for 0 < t, and 0 otherwise, and k is a constant
   b(t) = exp(-kt) for 0 < t, and 0 otherwise

e. a(t) = exp(-kt) for 0 < t, and 0 otherwise
   b(t) = -2 delta(t-2)


3. Convolve the following signals by differentiating one to reduce it to
impulses (such as in Fig. 13-7).

a(t) = 1 for 0 < t < 4, and 0 otherwise
b(t) = sin(2 pi t) for -1 < t < 1, and 0 otherwise  


4. Calculate the Fourier transform of the following signals.  Give your
answer in rectangular form.

a. x(t) = 1 for -1 < t < 1, and 0 otherwise. 
b. x(t) = t for -1 < t < 1, and 0 otherwise
c. x(t) = delta(t)
d. x(t) = exp(-abs(kt)), where abs is the absolute value and k is a constant


5. Which of the following signals has a larger 15th harmonic: a square wave,
a triangle wave, or a sawtooth wave?