Chapter 2 Frequency Domain Analysis of Signals and Systems

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Chapter 2 Frequency Domain Analysis of Signals and Systems

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Title: Chapter 2 Frequency Domain Analysis of Signals and Systems

1
Chapter 2Frequency Domain Analysis of Signals
and Systems
2
CONTENTS

Fourier Series
Fourier Transforms
Power and Energy
Sampling of Bandlimited Signals
Bandpass Signals

3
2.1 FOURIER SERIES

Theorem 2.1.1 Fourier Series Let the signal
x(t) be a periodic signal with period T0.If the
following conditions are satisfied
1. x(t) is absolutely integrable over its
period
2. The number of maxima and minima of x(t)
in each period is finite
3. The number of discontinuous of x(t) in
each period is finite
then x(t) can be expanded in terms of the
complex exponential signal as
where
for some arbitrary and

xn are called the Fourier series coefficients of
the signal x(t).
For all practice purpose,
From now on, we will use instead of
The quantity is called the
fundamental frequency of the signal x(t)
The Fourier series expansion can be expressed in
terms of angular frequency by
and

Discrete spectrum - we may represent
where
gives the magnitude of the nth
harmonic and gives its phase.

Example Let x(t) denote the periodic signal
depicted in Figure 2.2

where
is a rectangular pulse. Determine the Fourier
series expansion for this signal
7

Solution We first observe that the period of the
signal is T0 and

Therefore, we have

Superposition of

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2.2.1 Fourier Series for Real Signals

If the signal x(t) is a real signal, we have

For a real, periodic x(t), the positive and
negative coefficients are conjugates.
has even symmetry and has odd
symmetry with respect to the n0 axis.

We may let and

This relation is called the trigonometric Fourier
series expansion.

To obtain and , we have

From above equation, we obtain

There exists a third way to represent the Fourier
series expansion of a real signal. Noting that

we have

For a real periodic signal, we have three
alternative ways to represent Fourier series
expansion

The corresponding coefficients are obtained from

16
2.2 FOURIER TRANSFORMS

Fourier transform is the extension of Fourier
series to periodic and nonperiodic signals.
The signal are expressed in terms of complex
exponentials of various frequencies, but these
frequencies are not discrete.
The signal has a continuous spectrum as opposed
to a discrete spectrum.

Theorem 2.2.1 Fourier Transform If the signal
x(t) satisfies certain conditions known as
Dirichlet conditions, namely,
1. x(t) is absolutely integrable on the
real line, i.e.,

2. The number of maxima and minima of x(t) in any
finite interval on the real line is finite, 3.
The number of discontinuities of x(t) in any
finite interval on the real line is finite,
Then, the Fourier transform of x(t), defined by
And the original signal can be obtained from its
Fourier transform by
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Observations
X(f) is in general a complex function. The
function X(f) is sometimes referred to as the
spectrum of the signal x(t).
To denote that X(f) is the Fourier transform of
x(t), the following notation is frequently
employed
to denote that x(t) is the inverse Fourier
transform of X(f) , the following notation is
used
Sometimes the following notation is used as a
shorthand for both relations

The Fourier transform and the inverse Fourier
transform relations can be written as
On the other hand,
where is the unit impulse. From
above equation, we may have
or, in general
hence, the spectrum of is equal to
unity over all frequencies.

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Example 2.2.1 Determine the Fourier transform of
the signal . Solution We have
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The Fourier transform of .

22
Example 2.2.2 Find the Fourier transform of the
impulse signal . Solution The
Fourier transform can be obtained
by Similarly, from the relation We conclude
that
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The Fourier transform of .

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2.2.1 Fourier Transform of Real, Even, and Odd
Signals

The Fourier transform can be written in general
as
For real x(t),

Since cosine is an even function and sine is an
odd function, we see that, for real x(t), the
real part of X(f) is an even function of f and
the imaginary part is an odd function of f.
Therefore, we have
This is equivalent to the following relations

Magnitude and phase of the spectrum of a real
signal.

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2.2.2 Basic Properties of the Fourier Transform

Linearity Property Given signals and
with the Fourier transforms
The Fourier transform of
is

Duality Property
Time Shift Property A shift of in the time
origin causes a phase shift of in
the frequency domain.
Scaling Property For any real , we
have

Convolution Property If the signal and
both possess Fourier transforms, then
Modulation Property The Fourier transform of
is , and the Fourier
transform of
is

Parsevals Property If the Fourier transforms of
and are denoted by
and , respectively, then
Rayleighs Property If X(f) is the Fourier
transform of x(t), then

Autocorrelation Property The (time)
autocorrelation function of the signal x(t) is
denoted by and is defined by
The autocorrelation property states that
Differentiation Property The Fourier transform
of the derivative of a signal can be obtained
from the relation

Integration Property The Fourier transform of
the integral of a signal can be determined from
the relation
Moments Property If ,
then , the nth moment of x(t),
can be obtained from the relation

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2.2.3 Fourier Transform for Periodic Signals

Let x(t) be a periodic signal with period ,
satisfying the Dirichlet conditions. Let
denote the Fourier series coefficients
corresponding to this signal. Then
Since
we obtain

If we define the truncated signal as
we may have

By using the convolution theorem, we obtain
Comparing this result with
we conclude

Alternative way to find the Fourier series
coefficients, Given the periodic signal x(t), we
carry out the following steps to find
1. Find the truncated signal .
2. Determine the Fourier transform
of the
truncated signal.
3. Evaluate the Fourier transform of the
truncated signal
at , to obtain the nth
harmonic and multiply by

Example 2.2.3 Determine the Fourier series
coefficients of the signal x(t) shown in Figure
2.2.

Solution The truncated signal is and its
Fourier transform is Therefore,
39
2.3 POWER AND ENERGY

The energy content of a signal x(t), denoted by
, is defined as
and the power content of a signal is
A signal is energy-type if
and is power-type if
A signal cannot be both power- and energy-type
because for energy-type signals
and for power-type signals

All nonzero periodic signals with period
are power-type and have power
where is any arbitrary real number.

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2.3.1 Energy-Type Signal

For any energy-type signal x(t), we define the
autocorrelation function as
By setting , we obtain

This relation gives two methods for finding the
energy in a signal. One method uses x(t), the
time-domain representation of the signal, and the
other method uses X(f) , the frequency-domain
representation of the signal.
The energy spectral density of the signal x(t) is
defined by
The energy spectrum density represents the amount
of energy per hertz of bandwidth present in the
signal at various frequencies.

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2.3.2 Power-Type Signals

Define the time-average autocorrelation function
of the power-type signal x(t) as
The power content of the signal can be obtained
from

Define , the power-spectral density or
the power spectrum of the signal x(t) to be the
Fourier transform of the time-average
autocorrelation function
Now we may express the power content of the
signal x(t) in terms of , i.e.,

If a power-type signal x(t) is passed through a
filter with impulse response h(t), the output is
The time-average autocorrelation function for
the output signal is

By making a change of variables
and changing the order of integration we obtain
where in (a) we have use the definition of
and in (b) and (c) we have used the definition
of the convolution integral.

Taking the Fourier transform of both sides of
above equation, we obtain

For periodic signals, the time-average
autocorrelation function and the power spectral
density can be simplified considerably. Assume
that x(t) is a periodic signal with period
having the Fourier series coefficients .
The time-average autocorrelation function can be
expressed as

integrated over one period
49

If we substitute the Fourier series expansion of
the periodic signal in this relation, we obtain
Now using the fact that
we obtain
Time-average autocorrelation function of a
periodic signal is itself periodic with the same
period as the original signal, and its Fourier
series coefficients are magnitude squares of the
Fourier series coefficients of the original
signal.

The power spectral density of a periodic signal
The power content of a periodic signal
This relation is known as Rayleighs relation
for periodic signals.

If the periodic signal is passed through an LTI
system with frequency response H(f), the output
will be periodic. The power spectral density of
the output can be expressed as
The power content of the output signal is

52
2.4 SAMPLING OF BANDLIMITED SIGNALS

The sampling theorem is one of the most important
results in the analysis of signals.
Many modern signal processing techniques and the
digital communication methods are based on the
validity of this theorem.

is a slowly changed signal (small
bandwidth)
is a rapidly changed signal (large
bandwidth)
53

Sampling the signals and at
regular interval and ,
respectively results the sequence
and
To obtain an approximation of the original signal
we can use linear interpolation of the sampled
values.
It is obvious that the sampling interval
must be smaller than .
The sampling theorem states that
1. If the signal x(t) is bandlimited to W,
i.e., X(f)0 for
then it is sufficient to sample at
intervals
2. The original signal can be reconstructed
without
distortion from the samples as long as
the previous
condition is satisfied.

Theorem 2.4.1 Sampling theorem Let the signal
be bandlimited to W. Let x(t) be sampled at
multiples of some basic sampling interval ,
where . The sampled sequence
can be expressed as . Then it
is possible to reconstruct the original signal
x(t) from the sampled values by the
reconstruction formula
where is any arbitrary number that
satisfies
In the special case where
, the reconstruction relation simplifies to

Proof Let denote the result of sampling
the original signal by impulses at time
interval. Then
Now if take the Fourier transform of both
sides of the above equation and apply the dual of
the convolution theorem to the right-hand side,
we obtain

56
Low-pass filter
57

The relation shows that is a
replication of the Fourier transform of the
original signal at rate.
Now if , then the replicated
spectrum of x(t) overlaps, and the
reconstruction of the original signal is not
possible. This type of distortion that results
from under-sampling is known as aliasing error or
aliasing distortion.
If , no overlap occurs, and by
employing an appropriate filter we can
reconstruct the original signal back.

To reconstruct the original signal, it is
sufficient to filter the sampled signal by a low
pass filter with frequency response
1. for
2. for
For , the filter
can have any characteristic that makes its
implementation easy.
We may choose an ideal lowpass filter with
bandwidth where satisfies
, i.e.,
with this choice, we have

Taking inverse Fourier transform of both sides,
we obtain
We can reconstruct the original signal signal
perfectly, if we use sinc functions for
interpolation of the sampled values.
The sampling rate , which is
called the Nyquist sampling rate, is the minimum
sampling rate at which no aliasing occurs.

If sampling is done at the Nyquist rate, the only
choice for the reconstruction filter is an ideal
lowpass filter and

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2.5 BANDPASS SIGNAL

A bandpass signal x(t) whose frequency domain
X(f) has the characteristic
for
where

central frequency
62

Let , then it
can also be represented as
The term is called phasor
corresponds to x(t).
Define

Note that the frequency domain representation of
Z(f) is obtained by deleting the negative
frequencies from X(f) and multiplying the
positive frequencies by 2. By doing this, we have
From Table 2.1, we have

Duality Theorem
64

Using the convolution theorem
where
comparing the result with
We see that plays the same role as
is called the Hilbert transform of
.

By doing some frequency analysis on ,
we have
Hilbert transform is equivalent to a
phase shift for positive frequency and
phase shift for the negative frequency

The lowpass representation of the bandpass signal
x(t) can be represented by
and

is a low pass signal which is in general
a complex signal, i.e.,

in-phase
quadrature
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Substituting for and rewriting ,
we obtain
Equating the real and imaginary parts, we have

Define the envelope and phase of x(t) as
We can write
Comparing to phasor relation , we
can find that the only difference is that the
envelope and phase are both
time-varying functions.

envelope
phase
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The function z(t) can also be expressed as
From above equations, we have

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2.5.1 Transmission of Bandpass Signals through
Bandpass Systems

Let x(t) be a bandpass signal with center
frequency , and let h(t) be the impulse
response of an LTI system. Let y(t) be the
output of the system when driven by x(t).
In frequency domain we have Y(f)X(f)H(f). The
signal y(t) is also a bandpass signal.

By writing H(f) and X(f) in terms of their
lowpass equivalents, we obtain
Multiplying these two relations, we have
Finally, we obtain
or
To obtain y(t), we can carry out the convolution
at low frequency f0 , and then transform to
higher frequencies using

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