The signal
is bandlimited to 4 kHz. We want to sample it,
but it has been subjected to various signal processing
manipulations.
What sampling frequency (if any works) can be used
to sample the result of passing
through an RC highpass filter with
and
?
What sampling frequency (if any works) can be used to
sample the derivative of
?
The signal
has been modulated by an 8 kHz
sinusoid having an unknown phase: the resulting
signal is
, with
and
Can the modulated signal be sampled so that the
original signal can be recovered from
the modulated signal regardless of the phase value
? If so, show how and
find the smallest sampling rate that can be used; if not,
show why not.
Non-Standard Sampling
Using the properties of the Fourier series can ease
finding a signal's spectrum.
Suppose a signal
is periodic with period . If
represents the signal's Fourier series
coefficients, what are the Fourier series
coefficients of
?
Find the Fourier series of the signal
shown in [link].
Suppose this signal is used to sample a signal
bandlimited to
. Find an expression for and sketch the spectrum
of the sampled signal.
Does aliasing occur? If so, can a change in sampling
rate prevent aliasing;
if not, show how the signal can be
recovered from these samples.
Pulse Signal
A Different Sampling Scheme
A signal processing engineer from Texas
A&M claims to have developed an improved sampling
scheme. He multiplies the bandlimited signal by the
depicted periodic pulse signal to perform sampling ([link]).
Find the Fourier spectrum of this signal.
Will this scheme work? If so, how should
be related to the signal's bandwidth?
If not, why not?
Bandpass Sampling
The signal
has the indicated spectrum.
What is the minimum sampling rate for this signal
suggested by the Sampling Theorem?
Because of the particular structure of this
spectrum, one wonders whether a lower sampling rate
could be used. Show that this is indeed the case, and
find the system that reconstructs
from its samples.
Sampling Signals
If a signal is bandlimited to
Hz, we can sample it at
any rate
and recover the waveform exactly. This statement of the
Sampling Theorem can be taken to mean that all
information about the original signal can be extracted
from the samples. While true in principle, you do have
to be careful how you do so. In addition to the rms
value of a signal, an important aspect of a signal is
its peak value, which equals
.
Let
be a sinusoid having frequency
Hz. If we sample it
at precisely the Nyquist rate, how accurately do the
samples convey the sinusoid's amplitude? In other
words, find the worst case example.
How fast would you need to sample for the
amplitude estimate to be within 5% of the true
value?
Another issue in sampling is the inherent amplitude
quantization produced by A/D converters. Assume the
maximum voltage allowed by the converter is
volts and that it quantizes amplitudes to
bits.
We can express the quantized sample
as
, where
represents the quantization error at the
sample. Assuming the converter rounds, how large is
maximum quantization error?
We can
describe the quantization error as noise, with a
power proportional to the square of the maximum
error. What is the signal-to-noise ratio of the
quantization error for a full-range sinusoid?
Express your result in decibels.
Hardware Error
An A/D converter has a curious hardware problem:
Every other sampling pulse is half its normal amplitude
([link]).
Find the Fourier series for this signal.
Can this signal be used to sample a bandlimited signal
having highest frequency
?
Simple D/A Converter
Commercial digital-to-analog converters don't work this
way, but a simple circuit illustrates how they work.
Let's assume we have a
-bit converter. Thus, we
want to convert numbers having a
-bit representation into
a voltage proportional to that number. The first step
taken by our simple converter is to represent the number
by a sequence of pulses
occurring at multiples of a time interval
. The presence of a
pulse indicates a “1” in the corresponding
bit position, and pulse absence means a “0”
occurred. For a 4-bit converter, the number 13 has the
binary representation 1101
() and would be represented by the depicted
pulse sequence. Note that the pulse sequence is
“backwards” from the binary representation.
We'll see why that is.
This signal
serves as the input to a first-order RC lowpass filter.
We want to design the filter and the parameters and
so that
the output voltage at time
(for a 4-bit converter) is proportional to the
number. This combination of pulse creation and
filtering constitutes our simple D/A converter. The
requirements are
The voltage at time
should diminish by a factor of 2 the further the
pulse occurs from this time. In other words, the
voltage due to a pulse at
should be twice that of a pulse produced at
,
which in turn is twice that of a pulse at
,
etc.
The 4-bit D/A
converter must support a 10 kHz sampling
rate.
Show the circuit that works. How do the
converter's parameters change with sampling rate
and number of bits in the converter?
Discrete-Time Fourier Transforms
Find the Fourier transforms of the following sequences, where
is some sequence having Fourier transform
.
Spectra of Finite-Duration Signals
Find the indicated spectra for the following signals.
The discrete-time Fourier transform of
The discrete-time Fourier transform of
The discrete-time Fourier transform of
The length-8 DFT of the previous signal.
Just Whistlin'
Sammy loves to whistle and decides to record and analyze his whistling in lab.
He is a very good whistler; his whistle is a pure sinusoid that can be described by
.
To analyze the spectrum, he samples his recorded whistle with a sampling interval of
to obtain
.
Sammy (wisely) decides to analyze a few samples at a time, so he grabs 30 consecutive, but arbitrarily chosen, samples.
He calls this sequence
and realizes he can write it as
Did Sammy under- or over-sample his whistle?
What is the discrete-time Fourier transform of
and how does it depend on
?
How does the 32-point DFT of
depend on
?
Discrete-Time Filtering
We can find the input-output relation for a
discrete-time filter much more easily than for analog
filters. The key idea is that a sequence can be written
as a weighted linear combination of unit samples.
Show that
where
is the unit-sample.
If
denotes the unit-sample
response—the output of a discrete-time
linear, shift-invariant filter to a unit-sample
input—find an expression for the output.
In particular, assume our filter is FIR, with the
unit-sample response having duration
. If the input has duration , what is
the duration of the filter's output to this
signal?
Let the filter be a boxcar averager:
for
and zero otherwise.
Let the input be a pulse of unit height and duration
.
Find the filter's output when
,
an odd integer.
A Digital Filter
A digital
filter has the depicted unit-sample reponse.
What is the difference equation that defines this
filter's input-output relationship?
What is this filter's transfer function?
What is the filter's output when the input is
?
A Special Discrete-Time Filter
Consider a FIR filter governed by the difference equation
Find this filter's unit-sample response.
Find this filter's transfer function.
Characterize this transfer function
(i.e., what classic filter category
does it fall into).
Suppose we take a sequence and stretch it out by
a factor of three.
Sketch the sequence
for some example
. What is the filter's output to this
input? In particular, what is the output at the
indices where the input
is intentionally zero? Now how would you characterize this
system?
Simulating the Real World
Much
of physics is governed by differntial equations, and we
want to use signal processing methods to simulate physical
problems. The idea is to replace the derivative with a
discrete-time approximation and solve the resulting
differential equation. For example, suppose we have the
differential equation
and we approximate the derivative by
where essentially
amounts to a sampling interval.
What is the difference equation that must be
solved to approximate the differential equation?
When
, the unit step, what will be the simulated output?
Assuming
is a sinusoid, how should the sampling
interval be chosen so
that the approximation works well?
Derivatives
The derivative of a sequence makes little sense, but still, we can approximate it.
The digital filter described by the difference equation
resembles the derivative formula.
We want to explore how well it works.
Suppose the signal
is a sampled analog signal:
.
Under what conditions will the filter act like a differentiator?
In other words, when will
be proportional to
?
The DFT
Let's explore the DFT and its properties.
What is the
length- DFT of
length- boxcar
sequence, where
?
Consider the special case where
. Find the inverse DFT of the product of
the DFTs of two length-3 boxcars.
If we could use DFTs to perform linear filtering, it
should be true that the product of the input's
DFT and the unit-sample response's DFT equals
the output's DFT. So that you can use what you
just calculated, let the input be a boxcar signal
and the unit-sample response also be a boxcar. The
result of part (b) would then be the filter's
output if we could implement
the filter with length-4 DFTs. Does the actual
output of the boxcar-filter equal the result found
in the previous part?
What would you need to change so that the
product of the DFTs of the input and unit-sample
response in this case equaled the DFT of the
filtered output?
DSP Tricks
Sammy is faced with computing lots
of discrete Fourier transforms. He will, of course, use
the FFT algorithm, but he is behind schedule and needs
to get his results as quickly as possible. He gets the
idea of computing two transforms at
one time by computing the transform of
, where
and
are two real-valued signals of which he needs
to compute the spectra. The issue is whether he can retrieve
the individual DFTs from the result or not.
What will be the DFT
of this complex-valued signal in terms of
and
, the DFTs of the original signals?
Sammy's friend, an Aggie who knows some signal
processing, says that retrieving the wanted DFTs is easy:
“Just find the real and imaginary parts of
.” Show that this approach is too
simplistic.
While his friend's idea is not correct, it does
give him an idea. What approach will work?
Hint: Use the symmetry properties
of the DFT.
How does the number of computations change with
this approach? Will Sammy's idea ultimately lead to a
faster computation of the required DFTs?
Discrete Cosine Transform (DCT)
The discrete cosine transform of a
length- sequence is
defined to be
Note that the number of frequency terms is
:
.
Find the inverse DCT.
Does a Parseval's Theorem hold for the DCT?
You choose to transmit information about the signal
according to the DCT coefficients.
You could only send one, which one would you send?
A Digital Filter
A digital filter is described by the following
difference equation:
What is this filter's unit sample response?
What is this filter's transfer function?
What is this filter's output when the input is
?
Another Digital Filter
A digital filter is determined by the following difference equation.
Find this filter's unit sample response.
What is the filter's transfer function?
How would you characterize this filter (lowpass, highpass, special purpose, ...)?
Find the filter's output when the input is the sinusoid
.
In another case, the input sequence is zero for
, then becomes nonzero.
Sammy measures the output to be
.
Can his measurement be correct?
In other words, is there an input that can yield this output?
If so, find the input
that gives rise to this output.
If not, why not?
Yet Another Digital Filter
A filter has an input-output relationship given by the difference
equation
.
What is the filter's transfer function?
How would you characterize it?
What is the filter's output when the input equals
?
What is the filter's output
when the input is the depicted discrete-time
square wave ([link])?
A Digital Filter in the Frequency Domain
We have a filter with the transfer function
operating on the input signal
that yields the output
.
What is the filter's unit-sample response?
What is the discrete-Fourier transform of the output?
What is the time-domain expression for the output?
Digital Filters
A discrete-time system is governed by the difference equation
Find the transfer function for this system.
What is this system's output when the input is
?
If the output is observed to be
, then what is the input?
Digital Filtering
A digital filter has an input-output relationship
expressed by the difference equation
.
Plot the magnitude and phase of this filter's transfer
function.
What is this filter's output when
?
Detective Work
The signal
equals
.
Find the length-8 DFT (discrete Fourier transform) of
this signal.
You are told that when
served as the input to a linear FIR (finite impulse
response) filter, the output was
.
Is this statement true? If so, indicate why and
find the system's unit sample response; if not, show
why not.
A discrete-time, shift invariant, linear system produces
an output
when its input
equals a unit sample.
Find the difference equation governing the system.
Find the output when
.
How would you describe this system's function?
Time Reversal has Uses
A discrete-time system has transfer function
.
A signal
is passed through this system to yield the signal
.
The time-reversed signal
is then passed through the system to yield the
time-reversed output
.
What is the transfer function between
and
?
Removing “Hum”
The slang word “hum” represents power line waveforms
that creep into signals because of poor circuit
construction. Usually, the 60 Hz signal (and its
harmonics) are added to the desired signal. What we
seek are filters that can remove hum. In this problem,
the signal and the accompanying hum have been sampled;
we want to design a digital filter
for hum removal.
Find filter coefficients for the length-3 FIR filter
that can remove a sinusoid having
digital frequency
from its input.
Assuming the sampling rate is
to what analog frequency does
correspond?
A more general approach is to design a filter having a
frequency response magnitude
proportional to the absolute value of a cosine:
. In this way, not only can the fundamental but
also its first few harmonics be removed. Select the
parameter and the sampling
rate so that the frequencies at which the cosine equals zero
correspond to 60 Hz and its odd harmonics through the fifth.
Find the difference equation that defines this
filter.
Digital AM Receiver
Thinking that digital implementations are
always better, our clever engineer
wants to design a digital AM receiver. The receiver
would bandpass the received signal, pass the result
through an A/D converter, perform all the demodulation
with digital signal processing systems, and end with a
D/A converter to produce the analog message signal.
Assume in this problem that the carrier frequency is
always a large even multiple of the
message signal's bandwidth .
What is the smallest sampling rate that would be
needed?
Show the block diagram of the
least complex digital AM receiver.
Assuming the channel adds white noise and that
a -bit A/D converter
is used, what is the output's signal-to-noise
ratio?
DFTs
A problem on Samantha's homework asks for the
8-point DFT of the discrete-time
signal
.
What answer should Samantha obtain?
As a check, her group partner Sammy says that he
computed the inverse DFT of her answer and got
. Does Sammy's result mean that Samantha's
answer is wrong?
The homework problem says to lowpass-filter the
sequence by multiplying its DFT by
and then computing the inverse DFT. Will this
filtering algorithm work? If so, find the filtered
output; if not, why not?
Stock Market Data Processing
Because a trading week lasts five days, stock markets
frequently compute running averages each day over the
previous five trading days to smooth price fluctuations.
The technical stock analyst at the Buy-Lo--Sell-Hi
brokerage firm has heard that FFT filtering techniques
work better than any others (in terms of producing more
accurate averages).
What is the difference equation governing the
five-day averager for daily stock prices?
Design an efficient FFT-based filtering
algorithm for the broker. How much data should be
processed at once to produce an efficient algorithm?
What length transform should be used?
Is the analyst's information correct that FFT
techniques produce more accurate averages than any
others? Why or why not?
Echoes
Echoes not only occur in canyons, but also in auditoriums and telephone circuits.
In one situation where the echoed signal has been sampled, the input signal
emerges as
.
Find the difference equation of the system that models the production of echoes.
To simulate this echo system, ELEC 241 students are asked to write the most efficient (quickest) program that has the same input-output relationship.
Suppose the duration of
is 1,000 and that
,
,
, and
.
Half the class votes to just program the difference equation while the other half votes to program a frequency domain approach that exploits the speed of the FFT.
Because of the undecided vote, you must break the tie.
Which approach is more efficient and why?
Find the transfer function and difference equation of the system that suppresses the echoes.
In other words, with the echoed signal as the input, what system's output is the signal
?
Digital Filtering of Analog Signals
RU Electronics wants to develop a filter that would be
used in analog applications, but that is implemented
digitally. The filter is to operate on signals that
have a 10 kHz bandwidth, and will serve as a lowpass
filter.
What is the block diagram for your filter
implementation? Explicitly denote which components
are analog, which are digital (a computer performs
the task), and which interface between analog and
digital worlds.
What sampling rate must be used and how many
bits must be used in the A/D converter for the
acquired signal's signal-to-noise ratio to be at
least 60 dB? For this calculation, assume the
signal is a sinusoid.
If the filter is a length-128 FIR filter (the
duration of the filter's unit-sample response equals
128), should it be implemented in the time or
frequency domain?
Assuming
is the transfer function of the digital filter, what
is the transfer function of your system?
Signal Compression
Because of the slowness of the Internet, lossy signal
compression becomes important if you want signals to be
received quickly. An enterprising 241 student has
proposed a scheme based on frequency-domain processing.
First of all, he would section the signal into
length- blocks, and
compute its -point DFT.
He then would discard (zero the spectrum) at
half of the frequencies, quantize
them to -bits, and send
these over the network. The receiver would assemble the
transmitted spectrum and compute the inverse DFT, thus
reconstituting an -point
block.
At what frequencies should the spectrum be
zeroed to minimize the error in this lossy
compression scheme?
The nominal way to
represent a signal digitally is to use simple
-bit quantization of
the time-domain waveform. How long should a section
be in the proposed scheme so that the required
number of bits/sample is smaller than that nominally
required?
Assuming that effective
compression can be achieved, would the proposed
scheme yield satisfactory results?