EEL 5525 Practice Exam #1
Fall 2011
Lectures 17

Name: ____________________________________
ID: _______________________________________
Instructions:
Complete the exam on the space provided. When time is called, stop all work and follow the instructions provided. Any work that is not collected when called for will not be graded. The honor system will be strictly enforced.
Allowed
Open book
Open notes
Pocket calculators and laptops
Figures and tables
Internet access to Sakai (only)
Disallowed
Unauthorized Internet access
Cell phones
Q1: Sampling Theorem and Quantization
You are to analyze the audio recording and playback system shown below. The input audio frequency range is fÎ[0, 3.5] kHz. The listener’s hearing range is fÎ[0, 8] kHz. The ADC operates at the programmable sample rate of f_{s }= n8kHz, n an integer.

a) What is the lowest sampling frequency f_{s} that will insure that the original audio signal x(t) can be (theoretically) reconstructed from its timeseries samples x[k], without aliasing?
b) The room is presumed quiet and you begin recording at a sample rate of f_{s}=8k Sa/s. When played back you hear a 2k Hz “buzzing sound” in the captured signal. What is the expected minimum frequency of the extraneous tone that could have created this effect?
c) You decide to place an ideal analog lowpass antialiasing filter in front of the ADC. What should be the filter’s passband cutoff frequency?
d) The signed ±10V ADC provides an 8bit output with an input x(t)<10V. What is the ADC’s quantization step size?
e) The signed 8bit ADC’s output is sent to an accumulator that produces an output given by:
(not a MAC)
What is the minimum number of integer bits that must be assigned to the accumulator to insure overflowfree runtime performance?
Q.2:zTransforms
Table 1: Primitive Signals and their zTransform  
Timedomain  ztransform 
d[k]  1 
u[k]  z/(z–1) 
a^{k}u[k]  z/(z–a) 
ka^{k}u[k]  az/(z–a)^{2} 
You are studying a causal signal x[k] having a ztransform X(z)= (z+1)^{2}/(z1)(z0.5)^{2}. The signal has a Heaviside expansion given by:
X(z)= (z+1)^{2}/((z1)(z0.5)^{2} ) = A + Bz/(z1) + Cz/(z0.5) + Dz/(z0.5)^{2}.
Invert X(z) (Hint: Think X(z) = X(z)/z).
a) What is A?
b) What is B?
c) What is C?
d) What is D?
e) What is x[k]?
f) What is x[¥]?
g) What is x[0]?
3. Sampling and Data Conversion:
A real signal x(t) = sin(2p(10^{3})t) + sin(2p(6*10^{3})t) (f_{1}=1kHz, f_{2}=6kHz) is presented to the system shown below.
a What is the Nyquist sampling rate (Sa/s)?
b If x(t) is sampled at a rate f_{s}=8kHz, what is the reconstructed signal in the form y(t) = A sin(2pf_{1}t) + B sin(2pf_{2}t)? (Assume the quantizer is bypassed, that is let x[k]=y[k])
c The qunatizer is inplace. The resulting signed 8bit ADC having a ±8 volt dynamic range quantizes the input an analog signal bounded by x(t)£ 5 volts. What is the ADC’s quantization step size in volts/bit?
d What is the statistical quantization error in bits (i.e., how many fractional bits are statistically preserved)?
4: Sampling Theorem and Quantization]
The home edition of American Idol uses the recording system shown below. The ADC is sampled at a 12000 Sa/s rate. The human vocal input is assumed limited to 4 kHz.
a. The sample rate is chosen to be 12k Sa/s. To test the system, a handheld audio signal generator is placed near the microphone. The signal generator produces a sinusoid tone x(t)=sin(2pf_{0}t) where f_{0} = 8kHz. What is the reconstructed signal y(t)?
b. The signal generator’s frequency is set to f_{0} = 4 kHz but the gain on the electronic signal generator, used in Part 1.b, is set too high and produces a square wave x(t) = sign(sin(2pf_{0}t)) having a Fourier series representation given by:]
Assume that x(t) can be essentially model using only the 1^{st}, 3^{rd}, and 5^{th} harmonics having amplitudes a_{1}= 2/p, a_{3}= 2/3p, and a_{5}= 2/5p respectively, where f_{0} = 4 kHz and f_{s} = 12 kSa/s. What is the reconstructed output signal y(t)?
5: Discretetime system
Consider the noncausal discretetime systems shown below.
The difference equation that applies to the system shown on the left of the Figure shown above is .
a. – Is the system BIBO stable?
b. What is the difference equation the applies to the system shown on the right of the Figure shown above?
c. What are the system’s first 4 outputs if y[1]=0 (system atrest) and x[k]=u[k] (unit step)?
Fall 2011
Lectures 17
Name: ____________________________________
ID: _______________________________________
Instructions:
Complete the exam on the space provided. When time is called, stop all work and follow the instructions provided. Any work that is not collected when called for will not be graded. The honor system will be strictly enforced.
Allowed
Open book
Open notes
Pocket calculators and laptops
Figures and tables
Internet access to Sakai (only)
Disallowed
Unauthorized Internet access
Cell phones
Q1: Sampling Theorem and Quantization
You are to analyze the audio recording and playback system shown below. The input audio frequency range is fÎ[0, 3.5] kHz. The listener’s hearing range is fÎ[0, 8] kHz. The ADC operates at the programmable sample rate of f_{s }= n8kHz, n an integer.

 What is the lowest sampling frequency f_{s} that will insure that the original audio signal x(t) can be (theoretically) reconstructed from its timeseries samples x[k], without aliasing?
 The room is presumed quiet and you begin recording at a sample rate of f_{s}=8k Sa/s. When played back you hear a 2k Hz “buzzing sound” in the captured signal. What is the expected minimum frequency of the extraneous tone that could have created this effect?
 You decide to place an ideal analog lowpass antialiasing filter in front of the ADC. What should be the filter’s passband cutoff frequency?
 The signed ±10V ADC provides an 8bit output with an input x(t)<10V. What is the ADC’s quantization step size?
 The signed 8bit ADC’s output is sent to an accumulator that produces an output given by:
(not a MAC)
What is the minimum number of integer bits that must be assigned to the accumulator to insure overflowfree runtime performance?
Q.2: zTransforms
Table 1: Primitive Signals and their zTransform  
Timedomain  ztransform 
d[k]  1 
u[k]  z/(z–1) 
a^{k}u[k]  z/(z–a) 
ka^{k}u[k]  az/(z–a)^{2} 
You are studying a causal signal x[k] having a ztransform X(z)= (z+1)^{2}/(z1)(z0.5)^{2}. The signal has a Heaviside expansion given by:
X(z)= (z+1)^{2}/((z1)(z0.5)^{2} ) = A + Bz/(z1) + Cz/(z0.5) + Dz/(z0.5)^{2}.
Invert X(z) (Hint: Think X(z) = X(z)/z).
 What is A?
 What is B?
 c) What is C?
 d) What is D?
 e) What is x[k]?
 f) What is x[¥]?
 g) What is x[0]?
3. Sampling and Data Conversion:
A real signal x(t) = sin(2p(10^{3})t) + sin(2p(6*10^{3})t) (f_{1}=1kHz, f_{2}=6kHz) is presented to the system shown below.
a What is the Nyquist sampling rate (Sa/s)?
b If x(t) is sampled at a rate f_{s}=8kHz, what is the reconstructed signal in the form y(t) = A sin(2pf_{1}t) + B sin(2pf_{2}t)? (Assume the quantizer is bypassed, that is let x[k]=y[k])
c The qunatizer is inplace. The resulting signed 8bit ADC having a ±8 volt dynamic range quantizes the input an analog signal bounded by x(t)£ 5 volts. What is the ADC’s quantization step size in volts/bit?
d What is the statistical quantization error in bits (i.e., how many fractional bits are statistically preserved)?
4: Sampling Theorem and Quantization]
The home edition of American Idol uses the recording system shown below. The ADC is sampled at a 12000 Sa/s rate. The human vocal input is assumed limited to 4 kHz.
 The sample rate is chosen to be 12k Sa/s. To test the system, a handheld audio signal generator is placed near the microphone. The signal generator produces a sinusoid tone x(t)=sin(2pf_{0}t) where f_{0} = 8kHz. What is the reconstructed signal y(t)?
 The signal generator’s frequency is set to f_{0} = 4 kHz but the gain on the electronic signal generator, used in Part 1.b, is set too high and produces a square wave x(t) = sign(sin(2pf_{0}t)) having a Fourier series representation given by:]
Assume that x(t) can be essentially model using only the 1^{st}, 3^{rd}, and 5^{th} harmonics having amplitudes a_{1}= 2/p, a_{3}= 2/3p, and a_{5}= 2/5p respectively, where f_{0} = 4 kHz and f_{s} = 12 kSa/s. What is the reconstructed output signal y(t)?
5: Discretetime system
Consider the noncausal discretetime systems shown below.
The difference equation that applies to the system shown on the left of the Figure shown above is .
 – Is the system BIBO stable?
b. What is the difference equation the applies to the system shown on the right of the Figure shown above?
c. What are the system’s first 4 outputs if y[1]=0 (system atrest) and x[k]=u[k] (unit step)?