Q1: Sampling Theorem and Quantization
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 fs
= n8kHz, n an integer.
is the lowest sampling frequency fs
that will insure that the original audio signal x(t) can be
(theoretically) reconstructed from its time-series samples x[k], without aliasing?
room is presumed quiet and you begin recording at a sample rate of fs=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?
decide to place an ideal analog lowpass anti-aliasing filter in front of the
ADC. What should be the filter’s passband cut-off frequency?
signed±10V ADC provides an 8-bit output with an
input |x(t)|<10V. What is the ADC’s
quantization step size?
signed 8-bit ADC’s output is sent to an accumulator that produces an output
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is the minimum number of integer bits that must be assigned to the accumulator
to insure overflow-free run-time performance?
Table 1: Primitive Signals and their z-Transform
are studying a causal signal x[k]
having a z-transform X(z)= (z+1)2/(z-1)(z-0.5)2. The signal has a Heaviside expansion given by:
) = A + Bz/(z-1) + Cz/(z-0.5)
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?
3. Sampling and Data Conversion:
real signal x(t) = sin(2p(103)t)
(f1=1kHz, f2=6kHz) is presented to the system shown below.
What is the Nyquist sampling rate (Sa/s)?
If x(t) is sampled at a rate fs=8kHz, what is the reconstructed
signal in the form y(t) = A sin(2pf1t) + B sin(2pf2t)? (Assume the quantizer
is bypassed, that is let x[k]=y[k])
qunatizer is in-place. The resulting signed 8-bit 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?
What is the statistical quantization error in bits (i.e., how many fractional
bits are statistically preserved)?
4: Sampling Theorem and
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
a. The sample rate is chosen to be 12k
Sa/s. To test the system, a hand-held audio signal generator is placed near the
microphone. The signal generator produces
a sinusoid tone x(t)=sin(2pf0t)
where f0 = 8kHz. What is the
reconstructed signal y(t)?
.gif">b. The signal generator’s frequency is set to f0
= 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(2pf0t)) having a Fourier series
representation given by:]
Assume that x(t) can be essentially
model using only the 1st, 3rd, and 5th
harmonics having amplitudes a1= 2/p, a3= 2/3p, and a5= 2/5p respectively, where f0
= 4 kHz and fs = 12 kSa/s. What is the reconstructed output signal y(t)?
5: Discrete-time system
the non-causal discrete-time systems shown below.
difference equation that applies to the system shown on the left of the Figure shown
above is .gif">.
Is the system BIBO stable?
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 at-rest) and x[k]=u[k] (unit step)?