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

e. x[n-3]
f. y[n+1]
g. 2x[n+1]
h. -y[n-1]
i. x[n] + y[n] 
j. -2x[n-1] + 3y[n+2]
k. 3x[n+2] - 2y[n+2]
l. x[n] + x[n-2]
m. 2x[n] -3x[n-2] + 3y[n+1]


2. Sketch the following discrete signals for -8 < n < 8:

a. x[n] = sin(2 pi n / 8)
b. x[n] = cos(2 pi n / 4)
c. x[n] = sin(2 pi n / 2)
d. x[n] = cos(2 pi n / 2)
e. x[n] = n-3 if n > 2, 0 otherwise
f. x[n] = 1 if n < -3, 0 if 0 < n < 4, 5 otherwise


3. Sketch the following continuous signals for -8 < t < 8: 

a. x(t) = sin(2 pi t / 8)
b. x(t) = cos(2 pi t / 4)
c. x(t) = sin(2 pi t / 2)
d. x(t) = cos(2 pi t / 2)
e. x(t) = n-3 if t > 2, 0 otherwise
f. x(t) = 1 if t < -3, 0 if 0 < t < 4, 5 otherwise


4. Samples 0 to 11 of a signal have the value: 0, 2, 3, 4, 3, 2,-1, 0, -2,
-3, 2, 1.   Calculate, sketch and label:
a. The even and odd parts.
b. The interlaced decomposition.
c. The step decomposition.


5. Two continuous waveforms, b(t) and x(t) are defined by: 

b(t) =  1 for 0 < t < 2
        0 otherwise

x(t) = -1 for 1 < t < 2
        1 for 2 < t < 3
        4 for 3 < t < 4
        2 for 4 < t < 5
        0 otherwise

a. Sketch b(t) and x(t)
b. Show that x(t) can be decomposed into three scaled and shifted versions of
b(t).  That is, find: a1, a2, a3, s1, s2, s3, such that: x(t) = a1 b(t-s1) +
a2 b(t-s2) + a3 b(t-s3)
c. Sketch these three component signals.


6. Systems are proven to be linear by mathematically showing that they obey
the properties of additivity and homogeneity.  However, systems in the real
world are often only understood by empirical measurements.  That is, a
scientist or engineer places a test signal into the input, and looks at what
comes out. 
        
a. Is it possible to prove that a system is linear based on measurements of
the input and output alone, without knowing the formal mathematical 
relationship between the input and output?  Explain.  

b. Is if possible to prove that a system is nonlinear in this way? Explain.

To help you answer these questions, think about an electronics technician
testing a "black box" for being linear.  He does this by placing signals
into the input and observing the output.  However, the technician has
absolutely no information about what is "inside" the system.  For instance,
it might contain an evil demon trying to deceive the technician.  Or in
another case, it might contain a timer that scrambles the output once every
ten million years of operation. 



CHAPTER 6: CONVOLUTION

1. A system has an impulse response, h[n], given by:  1, 2, 2, 1, 0, -1, 0,
0, for the values of samples 0 to 7.  Calculate the output of the system in
response to the following input signals.  

a.  1, 0, 0, 0, 0, 0, 0, 0, 0
b. -3, 0, 0, 0, 0, 0, 0, 0, 0 
c.  0, 0, 1, 0, 0, 0, 0, 0, 0
d.  1, 0, 1, 0, 0, 0, 0, 1, 0
e.  3, 0,-1, 0, 0, 2, 0, 0, 0
f.  2,-1, 0, 0, 1, 0,-1, 0, 0 


2.   Adding zeros to the end of a signal is a common DSP technique called
"padding with zeros."  Use your results in the last problem to answer the
following: 

a. How would the output signals be changed if 5 additional samples, all with
a value of zero, were added to the end of the impulse response?  
b. How would the output signals be changed if 5 additional samples, all with
a value of zero, were added to the end of the input signals? 
c. How would the output signals be changed if 5 additional samples, all with
a value of zero, were added to the end of both the input signals and the
impulse response?       
d. Complete the following statement summarizing how "padding with zeros"
affects convolution: "When M zeros are added to either the input signal or
the impulse response, the only change to the output signal is [fill in the
blank].  


3. Two signals, x[n] and h[n], are defined by:

x[n]:  1, 0, 2, 3, 2, 1,-1,-2,-1, 0, 2, 3, 3, 2, 1, 1  (samples 0-15) 
h[n]:  1, 2, 3,-3,-2,-1  (samples 0-5)  

If y[n] = x[n]*h[n], use the input side algorithm to determine the
contribution to y[n] from the indicated sample:

a. x[2]
b. x[6]
c. x[9]


4. For the signals in the last problem, use the output side algorithm to
calculate the value of the following samples:  

a. y[8]
b. y[10]
c. y[3]
d. y[18]


5. Two signals, a[n] and b[n], are defined by:

a[n]:  1, 0, 0, 2, 1, 0 
b[n]:  0,-1,-2, 0, 0, 1

a. Calculate a[n]*b[n] by using an impulse decomposition on a[n], convolving
each of the components with b[n], and synthesizing (adding) the resulting
signals.  
b. Calculate a[n]*b[n] by using an impulse decomposition on b[n], convolving
each of the components with a[n], and synthesizing (adding) the resulting
signals.  
c. Do the results of these two methods agree? What property is demonstrated
in this problem?  


6. Calculate the convolution of the signal: h[n] = 1, 2, 3, 0, 0 with the 
indicated signals (assume each of the following run from sample 0 to 7).

a. x[n] =  delta[n]
b. x[n] = -5 delta[n-2] 
c. x[n] =  2 delta[n+1] - delta[n+1]
d. x[n] =  1, 2, 3, 0, 0 ...
e. x[n] = -n  for  0 < n < 5, and 0 otherwise  
f. x[n] =  2^(-n)
g. x[n] = sin(2 pi n)
h. x[n] = cos(2 pi n)
i. x[n] = sin (pi n)     


7. Calculate the convolution of the following signals (your answer will be
in the form of an equation): 

a. h[n] = delta[n], x[n] = delta[n]
b. h[n] = delta[n], x[n] = delta[n-k]
c. h[n] = delta[n-2], x[n] = delta[n-1] + delta[n+4]
d. h[n] = delta[n-1] + delta[n+1], x[n] = delta[n-a] + delta[n+b] 
e. h[n] = delta[n], x[n] = exp(-n)
f. h[n] = delta[n+2], x[n] = exp(n)
g. h[n] = delta[n-2], x[n] = exp(-n)
h. h[n] = exp(-n), x[n] = delta[n-2]
i. h[n] = delta[n] - delta[n-1], x[n] = exp(-n)


8. A financial expert receives daily reports on the value of a particular
stock. Each day he calculates the average value of the stock over the last 
30 days.  If this averaging were describe as a system: 

a. What are the input and output signals?
b. Is this system linear?
c. What is the impulse response of the system?
d. What practical purpose would this system be serving?
e. What would be the impulse response if the average was taken over M days? 


9. If the signal, x[n], has a value of zero over the interval:  A <= n <= B, 
and if signal, h[n], has a value of zero over the interval: C <= n <= D, then 
x[n]*h[n] must be zero over the interval, E <= n <= F.  Express the variables, 
E and F, in terms of: A, B, C, and D.  


10. Two signals, a[n] and b[n], each contain 6 points, as defined below. 
Calculate a[n]*b[n].

a[n]:  1,  0,  0, 2, 1, 0  
b[n]:  0, -1, -2, 0, 0, 1  

a. Where both signals run from sample 0 to 5
b. Where both signals run from sample 2 to 7
c. Where a[n] runs from sample 0 to 5, and b[n] runs from sample -3 to 2
d. Where a[n] runs from sample -10 to -5, and b[n] runs from sample -5 to 0 



CHAPTER 7: PROPERTIES OF CONVOLUTION
 
1. Classify the following signals as either casual or noncausal.

a. x[n] = delta[n]
b. x[n] = delta[n-2]
c. x[n] = delta[n-1] + delta[n+1]
d. x[n] = delta[n] -  5 delta[n-5]
e. x[n] = delta[n] + delta[n+5]
f. x[n] = delta[n-1] - delta[n-4] + delta[n-7]
g. x[n] = exp(-n)
h. x[n] = exp(-abs(n))  (where "abs" is the absolute value function)
i. x[n] = abs(n)
j. x[n] = n + abs(n)


2. Classify the signals in the last problem as either zero phase, linear
phase, or nonlinear phase.


3. The impulse responses of three linear systems are given below.  Calculate
the impulse response of the indicated combination

system A:  3, 2, 1, 0
system B:  0, 1,-1, 0
system C:  1, 1, 1, 1

a. The parallel combination of system A and system B.
b. The parallel combination of system A, system B, and system C. 
c. The cascade of System A and system B. 
d. The cascade of System B and system A.
e. The cascade of System A and system B, in parallel with system C.


4. System A is an "all pass" system, meaning that its output is identical to
its input.  System B is a low-pass filter that passes all frequencies below
the cutoff frequency without change, and blocks all frequencies above.  Call
the impulse response of system B, b[n]. 

a. What is the impulse response of system A?
b. How would the impulse response of system B need to be changed to make the
system have an inverted output (i.e., the same output, just changed in sign)?
c. If the two systems are arranged in parallel with added outputs, what is
the impulse response of the combination?
d. If the two systems are arranged in parallel, with the output of system B
subtracted from the output of system A, what is the impulse response of the
combination?
e. What kind of filter is the system in (d)?
f. Describe an algorithm for changing a low-pass filter kernel into a high-
pass filter kernel. 
g. Describe an algorithm for changing a high-pass filter kernel into a low-
pass filter kernel.  Is this exactly the same procedure as in (f)?
h. In this problem, system B has the ideal characteristic of passing certain
frequencies "without change."  How would the algorithm you described in (f)
be affected if the low-pass filter delayed (i.e., shifted) the output signal
by a small amount, relative to the input signal?    


5. From calculus, you know that the derivative and integral are inverse
operations; one undoes the effect of the other.  Prove that the first
difference and the running sum are also inverse operations.  That is, show
that the cascade of these two systems is identical to the delta function. 


6. Echoes are added to audio signals to make the listener "feel" that they
are in a particular size of room.  Assume that an audio signal is sampled at
44 kHz, and that sound propagates at 332 meters/second. In a "small" room, a
person stands about 3 meters from the walls; in a "large" room, the distance
increases to about 10 meters.

a. In a small room, how long is the delay between a person making a sound
and its echo from the walls. 
b. How many samples does this correspond to in the digital signal?
c. What is the impulse response of a digital system simulating this echo, if
the amplitude of the echo is 20%?
d. Repeat (a) to (c) for the large room.
e. In a real listening environment, each echo will also generate another
echo.  That is, each original sound will be heard over and over with
diminishing amplitude. How would the impulse response in (c) be modified to
account for these echoes of echoes?



CHAPTER 8: THE DISCRETE FOURIER TRANSFORM

1. A sinusoid at 1.7 kHz is digitized at 10,000 samples per second.  The
signal is passed through a 2048 point DFT, and converted to polar form. Draw
four sketches of the magnitude, one for each of the four ways that the
frequency domain's independent variable can be expressed. Be sure to
indicate the frequency symbol used, the range of values, the units, and at
what frequency the sinusoid appears. 


2. A peak appears at index number 19 when a 256 point DFT is taken of a
signal.  

a. What is the frequency of the peak expressed as a fraction of the sampling
rate? Do you need to know the actual sampling rate to answer this question?
b. What is the frequency of the peak expressed as a natural frequency?
c. What is the sampling rate if the peak corresponds to 21.5 kHz in the
analog signal?
d. What is the frequency of the sinusoid (in hertz) if the sampling rate is
100 kHz?    


3. Calculate, sketch and label the basis functions for an 8 point DFT. 


4. An 8 sample signal is given by: 20,21,22,23,24,25,26,27.  Its frequency
domain is given by: R0,R1,R2,R3,R4 and I0,I1,I2,I3,I4.  Write the
simultaneous equations relating the frequency domain and the time domain.
Use the numerical value for each point in the basis functions, for example,
use 0.7071, not sin(pi/4). How difficult would it be to solve these
equations?  


5. Using the signals in the last problem, write 10 equations for calculating
the 10 points in the frequency domain, using the correlation algorithm.
Solve these equations. 


6. The frequency domain of a signal is given by:

real part:  1, 2, 3, 3, 1,-2,-1, 1, 2
imag part:  0,-1,-2, 0, 0, 0, 2, 1, 0

a. What length of DFT does this correspond to? 
b. Calculate the amplitudes of the sine and cosine waves that comprise the
time domain signal.
c. What is the mean (average value) of the time domain signal?


7. You are told that the following signals are the frequency domain of a 32
point real DFT. Give two reasons why this is not possible.
     
real part:  1,2,3,4,5,6,7,8,7,6,5,4,3,2,1,0 
imag part:  8,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7 


8. Convert the following real and imaginary parts into polar form. Sketch a
diagram of each, such as Fig. 8-9.  In each case, state if the conversion
equation: phase = arctan(IP/RP), provides the correct answer without
additional steps: 

a. RP =  1, IP =  1
b. RP =  1, IP = -1
c. RP = -1, IP =  1
d. RP = -1, IP = -1
e. RP =  1, IP =  0
f. RP = -1, IP =  0
g. RP =  0, IP =  1