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.
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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:
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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.
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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.
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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)?
.gif">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:]
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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.
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The
difference equation that applies to the system shown on the left of the Figure shown
above is .gif">.
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)?