Chapter 2
Frequency Domain Analysis of
Signals and Systems
CONTENTS
•
•
•
•
•
Fourier Series
Fourier Transforms
Power and Energy
Sampling of Bandlimited Signals
Bandpass Signals
1
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
∫
T0
0
| x (t ) |dt < ∞
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
∞
x± (t ) =
∑x e
n
j 2π
n
t
T0
n = −∞
where
n
xn =
for some arbitrary
α
and
− j 2π t
1 α +T0
T0
x
t
e
dt
(
)
∫
α
T0
x (t )
if x(t) is continous at t
⎧
x± (t ) = ⎨
+
−
⎩( x(t ) + x(t )) / 2 if x(t) is discontinous at t
•  xn are called the Fourier series coefficients of the signal
x± (t ) = x (t )
x(t).
•  For all practice purpose, x (t )
x± (t )
•  From now on, fwe
will use
instead of
0 = 1 / T0
•  The quantity
is called the fundamental frequency
of the signal x(t)
ω0 = 2πf 0 can be expressed in terms of
•  The Fourier series expansion
angular frequency α +2π / ω by
ω
xn = 0 ∫
x(t )e − jnω t dt
2π α
0
and
x (t ) =
∞
∑x e
n
0
jnω0 t
n = −∞
2
•  To obtain an and bn , we have
n
− j 2π t
a − jbn 1 α +T0
xn = n
= ∫ x(t )e T0 dt
2
T0 α
=
⎛
⎛
1 α +T0
n ⎞
j α +T0
n ⎞
x(t ) cos⎜⎜ 2π t ⎟⎟dt − ∫ x(t ) sin ⎜⎜ 2π t ⎟⎟dt
∫
α
α
T0
T0 ⎠
T0
⎝ T
⎝ 
0 ⎠

 



an
2
jbn
2
•  From above equation, we obtain
an =
⎛
2 α +T0
n ⎞
x(t ) cos⎜⎜ 2π t ⎟⎟dt
T0 ∫α
T
⎝
0 ⎠
bn =
⎛
2 α +T0
n ⎞
x(t ) sin ⎜⎜ 2π t ⎟⎟dt
∫
α
T0
⎝ T0 ⎠
•  There exists a third way to represent the Fourier series
expansion of a real signal. Noting that
xn e
we have
j 2π
n
t
T0
+ x− n e
− j 2π
n
t
T0
⎞
⎛
n
= 2 xn cos⎜⎜ 2π t + ∠xn ⎟⎟
⎠
⎝ T0
∞
⎞
⎛
n
x(t ) = x0 + 2∑ xn cos⎜⎜ 2π t + ∠xn ⎟⎟
T
n =1
⎠
⎝
0
•  For a real periodic signal, we have three alternative ways
to represent Fourier series expansion
x(t ) =
∞
∑ xn e
j 2π
n
t
T0
n = −∞
=
⎛
⎛
a0 ∞ ⎡
n ⎞
n ⎞⎤
+ ∑ ⎢an cos⎜⎜ 2π t ⎟⎟ + bn sin ⎜⎜ 2π t ⎟⎟⎥
2 n=1 ⎣
⎝ T0 ⎠
⎝ T0 ⎠⎦
∞
⎛
⎞
n
= x0 + 2∑ xn cos⎜⎜ 2π t + ∠xn ⎟⎟
n =1
⎝ T0
⎠
7
•  The corresponding coefficients are obtained from
1
xn =
T0
∫α
an =
2
T0
∫α
bn =
2
T0
∫α
α +T0
x(t )e
− j 2π
n
t
T0
dt =
an
b
+j n
2
2
α +T0
⎛
n ⎞
x(t ) cos⎜⎜ 2π t ⎟⎟dt
⎝ T0 ⎠
α +T0
⎛
n ⎞
x(t ) sin ⎜⎜ 2π t ⎟⎟dt
⎝ T0 ⎠
1 2
an + bn2
2
⎛ b ⎞
∠xn = − arctan⎜⎜ n ⎟⎟
⎝ an ⎠
xn =
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.
8
•  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.,
∫
∞
x(t ) dt < ∞
−∞
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
∞
− j 2πft
X ( f ) = ∫ x(t )e
dt
−∞
And the original signal can be obtained from its Fourier transform by
∞
x± (t ) = ∫ X ( f )e j 2πft df
−∞
•  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
X ( f ) = F [ x(t )]
to denote that x(t) is the inverse Fourier transform of X(f) , the
following notation is used
x(t ) = F −1[ X ( f )]
Sometimes the following notation is used as a shorthand for both
relations
x(t ) ⇔ X ( f )
9
–  The Fourier transform and the inverse Fourier transform relations
can be written as
∞
∞
x(t ) = ∫ ⎡ ∫ x(τ )e − j 2πfτ dτ ⎤e j 2πft df
⎥⎦
− ∞ ⎢
−
⎣ ∞
∞
∞
= ∫ ⎡ ∫ e j 2πf ( t −τ ) df ⎤x(τ )dτ
⎢
⎥⎦
− ∞ ⎣ − ∞
On the other hand,
∞
x(t ) = ∫ δ (t − τ )x(τ )dτ
−∞
where δ (t ) is the unit impulse. From above equation, we may
have
∞
j 2πf ( t −τ )
δ (t − τ ) = ∫ e
−∞
df
or, in general
∞
δ (t ) = ∫ e j 2πft df
−∞
hence, the spectrum of δ (t ) is equal to unity over all frequencies.
Example 2.2.1: Determine the Fourier transform of the
signal Π (t ).
Solution: We have
∞
F [Π (t )] = Π (t )e − j 2πft dt
∫
−∞
=
∫
1
2
1
−
2
Π (t )e − j 2πft dt
1
e − jπf − e jπf
− j 2πf
sin(πf )
=
πf
= sinc(f)
=
[
]
10
2.2.2 Basic Properties of the
Fourier Transform
•  Linearity Property: Given signals x1 (t ) and x2 (t ) with
the Fourier transforms
F [ x1 (t )] = X 1 ( f )
F [ x2 (t )] = X 2 ( f )
The Fourier transform of αx1 (t ) + βx2 (t ) is
F[αx1 (t ) + βx2 (t )] = αX1 ( f ) + βX 2 ( f )
•  Duality Property:
If X ( f ) = F[ x(t )] then x( f ) = F[ X (−t )] and x(− f ) = F [ X (t )]
•  Time Shift Property: A shift of t0 in the time origin
causes a phase shift of − 2πft0 in the frequency domain.
F [ x(t − t0 )] = e − j 2πft0 F [ x(t )]
•  Scaling Property: For any real a ≠ 0 , we have
F [ x(at )] =
1 ⎛ f ⎞
X ⎜ ⎟
a ⎝ a ⎠
14
•  Convolution Property: If the signal x (t ) and y (t ) both
possess Fourier transforms, then
F[ x(t ) ∗ y(t )] = F[ x(t )] ⋅ F[ y(t )] = X ( f ) ⋅ Y ( f )
j 2πf t
•  Modulation Property: The Fourier transform of x(t )e 0
is X ( f − f 0) , and the Fourier transform of
x(t ) cos(2πf 0t )
is
1
1
X ( f − f0 ) + X ( f + f0 )
2
2
•  Parseval’s Property: If the Fourier transforms of x (t )
and y (t ) are denoted by X ( f ) and Y ( f ) , respectively,
then
∞
∞
x(t ) y * (t )dt = X ( f )Y * ( f )df
∫
∫
−∞
−∞
•  Rayleigh’s Property: If X(f) is the Fourier transform of
x(t), then
∞
∫
2
x (t ) dt =
−∞
∞
2
∫ X ( f ) df
−∞
15
•  Autocorrelation Property: The (time) autocorrelation
function of the signal x(t) is denoted by Rx (τ ) and is
defined by
Rx (τ ) =
∞
*
∫ x(t) x (t − τ )dt
−∞
The autocorrelation property states that
F [ R x (τ )] = X ( f )
2
•  Differentiation Property: The Fourier transform of the
derivative of a signal can be obtained from the relation
⎡ d
⎤
F ⎢ x(t )⎥ = j 2πfX ( f )
dt
⎣
⎦
•  Integration Property: The Fourier transform of the
integral of a signal can be determined from the relation
⎡ ∞
⎤ X ( f ) 1
+ X (0)δ ( f )
F ⎢ x(τ )dτ ⎥ =
⎣ −∞
⎦ j 2πf 2
∫
∞
n
•  Moments Property: If F [ x(t )] = X ( f ) , then ∫−∞ t x(t )dt ,
the nth moment of x(t), can be obtained from the relation
n
n
⎛ j ⎞ d
t n x (t )dt = ⎜ ⎟
X( f )
n
−∞
⎝ 2π ⎠ df
f =0
∞
∫
16
•  If we define the truncated signal xT0 (t ) as
T
T
⎧⎪
x (t ) − 0 < t ≤ 0
xT0 (t ) = ⎨
2
2
⎪⎩ 0
otherwise
we may have
∞
x (t ) =
∑x
T0
(t − nT0 )
n = −∞
∞
= xT0 (t ) ∗
∑δ (t − nT )
0
n = −∞
•  By using the convolution theorem, we obtain
⎡ 1
X ( f ) = X T0 ( f ) ⎢
⎢⎣ T0
∞
∑
n = −∞
⎛
δ ⎜⎜ f −
⎝
n ⎞⎤
⎟⎥
T0 ⎟⎠⎥⎦
∞
=
⎛
1
n ⎞
X T0 ( f )δ ⎜⎜ f − ⎟⎟
T0 n = −∞
T0 ⎠
⎝
=
⎛ n ⎞ ⎛
1
n ⎞
X T ⎜ ⎟δ ⎜ f − ⎟⎟
T0 n = −∞ 0 ⎜⎝ T0 ⎟⎠ ⎜⎝
T0 ⎠
∑
∞
∑
Comparing this result with
∞
X( f ) =
∑ x δ ⎜⎜⎝ f − Tn ⎟⎟⎠
⎛
n = −∞
we conclude
xn =
⎞
n
0
⎛ n ⎞
1
X T0 ⎜⎜ ⎟⎟
T0
⎝ T0 ⎠
18
2.3 POWER AND ENERGY
•  The energy content of a signal x(t), denoted by E x , is
defined as
∞
Ex =
2
∫ x(t) dt
−∞
and the power content of a signal is
T
1 2
2
Px = lim
x(t ) dt
T
T →∞ T −
2
•  A signal is energy-type if Ex < ∞ and is power-type
if 0 < Px < ∞
•  A signal cannot be both power- and energy-type because
for energy-type signals Px = 0 and for power-type
signals Ex = ∞
∫
•  All nonzero periodic signals with period T0 are powertype and have power
Px =
1
T0
∫
α +T0
α
2
x (t ) dt
where α is any arbitrary real number.
20
2.3.1 Energy-Type Signal
•  For any energy-type signal x(t), we define the
autocorrelation function Rx (τ ) as
Rx (τ ) = x (τ ) ∗ x* ( −τ )
∫
=∫
=
∞
−∞
∞
−∞
x (t ) x* (t − τ )dt
x (t + τ ) x* (t )dt
•  By setting τ = 0 , we obtain
Rx (τ ) =
∫
=∫
∞
2
x (t ) dt
−∞
∞
2
X ( f ) df
−∞
= Ex
•  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
Gx ( f ) = F[ Rx (τ )] = X ( f )
2
•  The energy spectrum density represents the amount of
energy per hertz of bandwidth present in the signal at
various frequencies.
21
2.3.2 Power-Type Signals
•  Define the time-average autocorrelation function of the
power-type signal x(t) as
T
1
Rx (τ ) = lim ∫ 2T x(t ) x* (t − τ )dt
T →∞ T −
2
•  The power content of the signal can be obtained from
1
T →∞ T
Px = lim
T
2
T
−
2
∫
2
x(t ) dt
= Rx (0)
•  Define S x ( f ), the power-spectral density or the power
spectrum of the signal x(t) to be the Fourier transform of
the time-average autocorrelation function
S x ( f ) = F [ Rx (τ )]
•  Now we may express the power content of the signal x(t)
in terms of S x ( f ) , i.e.,
Px = Rx (0)
=
∫
∞
−∞
S x ( f )df
22
•  If we substitute the Fourier series expansion of the periodic
signal in this relation, we obtain
Rx (τ ) =
1
T0
T0
∞
∞
*
2
n m
T
− 0
2 n = −∞ m = −∞
∫ ∑∑
xx e
j 2π
m
τ
T0
e
j 2π
n −m
t
T0
dt
•  Now using the fact that
1
T0
T0
2
T
− 0
2
∫
e
j 2π
n −m
t
T0
we obtain
⎧1 n = m
dt = ⎨
⎩0 n ≠ m
≡ δ m ,n
∞
Rx (τ ) =
∑x
2
n
j 2π
e
n
τ
T0
n = −∞
•  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
∞
2 ⎛
n ⎞
S x ( f ) = F [ Rx (τ )] = ∑ xn δ ⎜⎜ f − ⎟⎟
T0 ⎠
n = −∞
⎝
•  The power content of a periodic signal
Px =
∫
∞
−∞
S x ( f )df =
∞
=
∑x
∞
∞
∫ ∑x
−∞
n
n = −∞
2
⎛
δ ⎜⎜ f −
⎝
n ⎞
⎟df
T0 ⎟⎠
2
n
n = −∞
This relation is known as Rayleigh’s relation for periodic
signals.
25
•  Sampling the signals x1 (t ) and x1 (t ) at regular interval T1
and T2 , respectively results the sequence x1 (nT1 ) and
x2 (nT2 ).
•  To obtain an approximation of the original signal we can
use linear interpolation of the sampled values.
•  It is obvious that the sampling interval T2 must be smaller
than T1 .
•  The sampling theorem states that
1. If the signal x(t) is bandlimited to W, i.e., X(f)=0 for f ≥ W
then it is sufficient to sample at intervals Ts = 1 /(2W ).
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 Ts , where Ts ≤ 1 /(2W ) . The
sampled sequence can be expressed as {x(nTs )}∞n=−∞ . Then
it is possible to reconstruct the original signal x(t) from the
sampled values by the reconstruction formula
∞
x (t ) =
∑ 2W ' T x(nT )sinc[2W' (t − nT )]
s
s
s
n = −∞
where W ' is any arbitrary number that satisfies
1
W ≤ W '≤ −W
Ts
In the special case where Ts = 1 /(2W ) , the reconstruction
relation simplifies to
∞
⎛ t
⎞ ∞ ⎛ n ⎞
⎡ ⎛ n ⎞⎤
x(t ) = ∑ x(nTs )sinc⎜⎜ − n ⎟⎟ = ∑ x⎜
⎟⎥
⎟sinc⎢2W ⎜ t⎣ ⎝ 2W ⎠⎦
n = −∞
⎝ Ts
⎠ n =−∞ ⎝ 2W ⎠
27
•  The relation shows that X δ ( f ) is a replication of the
Fourier transform of the original signal at 1 / Ts rate.
•  Now if Ts > 1 /(2W ) , 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 undersampling is known as aliasing error or aliasing distortion.
•  If Ts ≤ 1 /(2W ) , 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. H ( f ) = Ts for f < W
1
2. H ( f ) = 0 for f ≥ − W
Ts
1
For W ≤ f < − W , the filter can have any characteristic
Ts
that makes its implementation
easy.
•  We may choose an ideal lowpass filter with bandwidth W '
where W ' satisfies W ≤ W ' < 1 − W , i.e.,
Ts
⎛ f ⎞
H ( f ) = Ts Π ⎜
⎟
⎝ 2W ' ⎠
with this choice, we have
⎛ f ⎞
X ( f ) = X δ( f )Ts Π⎜
⎟
⎝ 2W ' ⎠
29
•  Taking inverse Fourier transform of both sides, we obtain
x (t ) = xδ (t ) ∗ 2W ' Ts sinc (2W ' t )
⎛ ∞
⎞
= ⎜⎜
x (nTs )δ (t − nTs ) ⎟⎟ ∗ 2W ' Ts sinc (2W ' t )
⎝ n = −∞
⎠
∑
∞
=
∑ 2W ' T x(nT )sinc(2W ' (t − nT ))
s
s
s
n = −∞
•  We can reconstruct the original signal signal perfectly, if
we use sinc functions for interpolation of the sampled
values.
•  The sampling rate f s = 1 /(2W ) , 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
W ' = W = 1 /(2Ts )
∞
x (t ) =
n
∑ x⎛⎜⎝ 2W ⎞⎟⎠sinc(2Wt − n )
n = −∞
⎛ t
∞
=
∑ x(nT )sinc⎜⎜⎝ T
s
n = −∞
s
⎞
− n ⎟⎟
⎠
30
•  By writing H(f) and X(f) in terms of their lowpass
equivalents, we obtain
X l ( f ) = 2u−1 ( f + f 0 ) X ( f + f 0 )
H l ( f ) = 2u−1 ( f + f 0 ) H ( f + f 0 )
Multiplying these two relations, we have
X l ( f ) H l ( f ) = 4u−1 ( f + f 0 ) X ( f + f 0 ) H ( f + f 0 )
Finally, we obtain
1
Yl ( f ) = X l ( f ) H l ( f )
2
or
1
yl (t ) = xl (t ) ∗ hl (t )
2
•  To obtain y(t), we can carry out the convolution at low
frequency f0 , and then transform to higher frequencies
using
y (t ) = Re yl (t )e j 2πf0t
[
]
37