Digital Signal Processing
Frequency Analysis of Signals (2)
Moslem Amiri, V´aclav Pˇrenosil
Embedded Systems Laboratory
Faculty of Informatics, Masaryk University
Brno, Czech Republic
amiri@mail.muni.cz
prenosil@fi.muni.cz
April, 2012
The Concept of Bandwidth
Low-frequency signal
A power signal (or energy signal) whose power density spectrum (or
energy density spectrum) is concentrated about zero frequency
Figure 1: Low-frequency signal.
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The Concept of Bandwidth
High-frequency signal
A power signal (or energy signal) whose power density spectrum (or
energy density spectrum) is concentrated at high frequencies
Figure 2: High-frequency signal.
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The Concept of Bandwidth
Medium-frequency signal or bandpass signal
A power signal (or energy signal) whose power density spectrum (or
energy density spectrum) is concentrated somewhere in broad frequency
range between low frequencies and high frequencies
Figure 3: Medium-frequency signal.
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The Concept of Bandwidth
Bandwidth of a signal
To express quantitatively the range of frequencies over which power or
energy density spectrum is concentrated
E.g., if a continuous-time signal has 95% of its power (or energy)
density spectrum concentrated in F1 ≤ F ≤ F2, then 95% bandwidth of
signal is F2 − F1
A bandpass signal is narrowband if its bandwidth F2 − F1 is much
smaller than median frequency (F2 + F1)/2
Otherwise, it is wideband
A signal is bandlimited if its spectrum is zero outside frequency range
|F| ≥ B
A periodic continuous-time signal xp(t) is bandlimited if its Fourier
coefficients ck = 0 for |k| > M (M is some positive integer)
A continuous-time finite-energy signal x(t) is bandlimited if its Fourier
transform X(F) = 0 for |F| > B
A periodic discrete-time signal with fundamental period N is
periodically bandlimited if Fourier coefficients ck = 0 for k0 < |k| < N
A discrete-time finite-energy signal x(n) is (periodically) bandlimited if
|X(ω)| = 0 for ω0 < |ω| < π
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The Concept of Bandwidth
Exploiting duality between frequency and time domains, a signal x(t)
is time-limited if
x(t) = 0, |t| > τ
If a signal is periodic with period Tp, it is periodically time-limited if
xp(t) = 0, τ < |t| < Tp/2
A discrete-time signal x(n) of finite duration (x(n) = 0, |n| > N) is also
time-limited
When a signal is periodic with fundamental period N, it is periodically
time-limited if
x(n) = 0, n0 < |n| < N
No signal can be time-limited and bandlimited simultaneously
A reciprocal relationship exists between time duration and frequency
duration of a signal
The narrower the pulse becomes in time domain, the larger the
bandwidth of signal becomes
Consequently, product of time duration and bandwidth of a signal is
fixed
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Frequency-Domain and Time-Domain Signal Properties
There are two time-domain characteristics that determine type of
signal spectrum we obtain
1 Whether time variable is continuous or discrete
2 Whether signal is periodic or aperiodic
Summary of results of previous sections
Continuous-time signals have aperiodic spectra
Because complex exponential exp(j2πFt) is a function of continuous
variable t, and hence it is not periodic in F
Thus frequency range extends from F = 0 to F = ∞
Discrete-time signals have periodic spectra
Both Fourier series and transform are periodic here with period ω = 2π
Frequency range is finite and extends from ω = −π to ω = π, where
ω = π corresponds to the highest possible rate of oscillation
Periodic signals have discrete spectra
Periodic signals are described by means of Fourier series
Fourier series coefficients provide lines that constitute discrete spectrum
Line spacing is ∆F = 1/Tp for continuous-time periodic signals and
∆f = 1/N for discrete-time signals
Aperiodic finite energy signals have continuous spectra
Because X(F) and X(ω) are functions of exp(j2πFt) and exp(jωn),
respectively, which are continuous functions of F and ω
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Frequency-Domain and Time-Domain Signal Properties
Figure 4: Summary of analysis and synthesis formulas.
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Frequency-Domain and Time-Domain Signal Properties
Periodicity with ”period” α in one domain automatically implies
discretization with ”spacing” of 1/α in the other domain, and vice
versa
”Period” in frequency domain means frequency range
”Spacing” in time domain is sampling period T
Line spacing in frequency domain is ∆F
α = Tp −→ 1/α = 1/Tp = ∆F
α = N −→ ∆f = 1/N
α = Fs −→ T = 1/Fs
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Properties of Fourier Transform for Discrete-Time Signals
Direct transform (analysis equation)
X(ω) ≡ F{x(n)} =
∞
n=−∞
x(n)e−jωn
Inverse transform (synthesis equation)
x(n) ≡ F−1
{X(ω)} =
1
2π 2π
X(ω)ejωn
dω
x(n) and X(ω) are a Fourier transform pair
x(n)
F
←→ X(ω)
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Properties of Fourier Transform for Discrete-Time Signals
Suppose both x(n) and X(ω) are complex-valued
x(n) = xR(n) + jxI (n) (1)
X(ω) = XR(ω) + jXI (ω) (2)
Putting (1) and e−jω = cos ω − j sin ω in X(ω) = ∞
n=−∞ x(n)e−jωn
XR(ω) =
∞
n=−∞
[xR(n) cos ωn + xI (n) sin ωn] (3)
XI (ω) = −
∞
n=−∞
[xR(n) sin ωn − xI (n) cos ωn] (4)
Putting (2) and ejω = cos ω + j sin ω in x(n) = 1
2π 2π X(ω)ejωndω
xR(n) =
1
2π 2π
[XR(ω) cos ωn − XI (ω) sin ωn]dω (5)
xI (n) =
1
2π 2π
[XR(ω) sin ωn + XI (ω) cos ωn]dω (6)
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Properties of Fourier Transform for Discrete-Time Signals
Real signals. If x(n) is real, then xR(n) = x(n) and xI (n) = 0
Hence (3) and (4) reduce to
XR (ω) =
∞
n=−∞
x(n) cos ωn and XI (ω) = −
∞
n=−∞
x(n) sin ωn
Since cos(−ωn) = cos ωn and sin(−ωn) = − sin ωn
XR (−ω) = XR (ω) and XI (−ω) = −XI (ω)
X∗
(ω) = X(−ω)
Magnitude and phase spectra for real signals
|X(ω)| = X2
R (ω) + X2
I (ω) and X|ω| = tan−1 XI (ω)
XR (ω)
|X(ω)| = |X(−ω)| and X(−ω) = − X(ω)
For inverse transform of a real-valued signal (x(n) = xR (n)), (5) implies
x(n) =
1
2π 2π
[XR (ω) cos ωn − XI (ω) sin ωn]dω
Since both XR (ω) cos ωn and XI (ω) sin ωn are even functions of ω
x(n) =
1
π
π
0
[XR (ω) cos ωn − XI (ω) sin ωn]dω
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Properties of Fourier Transform for Discrete-Time Signals
Real and even signals. If x(n) is real and even (x(−n) = x(n)), then
x(n) cos ωn is even and x(n) sin ωn is odd
XR(ω) =
∞
n=−∞
x(n) cos ωn −→ XR(ω) = x(0) + 2
∞
n=1
x(n) cos ωn (even)
XI (ω) = −
∞
n=−∞
x(n) sin ωn −→ XI (ω) = 0
x(n) =
1
π
π
0
[XR(ω) cos ωn − XI (ω) sin ωn]dω −→
x(n) =
1
π
π
0
XR(ω) cos ωndω
Thus spectra for real and even signals are
Real-valued
Even functions of ω
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Properties of Fourier Transform for Discrete-Time Signals
Real and odd signals. If x(n) is real and odd (x(−n) = −x(n)), then
x(n) cos ωn is odd and x(n) sin ωn is even
XR(ω) =
∞
n=−∞
x(n) cos ωn −→ XR(ω) = 0
XI (ω) = −
∞
n=−∞
x(n) sin ωn −→ XI (ω) = −2
∞
n=1
x(n) sin ωn (odd)
x(n) =
1
π
π
0
[XR(ω) cos ωn − XI (ω) sin ωn]dω −→
x(n) = −
1
π
π
0
XI (ω) sin ωndω
Thus spectra for real-valued odd signals are
Purely imaginary-valued
Odd functions of ω
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Properties of Fourier Transform for Discrete-Time Signals
Example
Determine and sketch XR(ω), XI (ω), |X(ω)|, and X(ω) for
X(ω) =
1
1 − ae−jω
, −1 < a < 1
Multiplying both numerator and denominator by complex conjugate of
denominator
X(ω) =
1 − aejω
(1 − ae−jω)(1 − aejω)
=
1 − a cos ω − ja sin ω
1 − 2a cos ω + a2
XR (ω) =
1 − a cos ω
1 − 2a cos ω + a2
and XI (ω) = −
a sin ω
1 − 2a cos ω + a2
|X(ω)| = X2
R (ω) + X2
I (ω) =
1
√
1 − 2a cos ω + a2
X|ω| = tan−1 XI (ω)
XR (ω)
= − tan−1 a sin ω
1 − a cos ω
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Properties of Fourier Transform for Discrete-Time Signals
Figure 5: Spectra of the transform in Example for a = 0.8; all symmetry
properties for the spectra of real signals apply to this case.
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Properties of Fourier Transform for Discrete-Time Signals
Example
Determine Fourier transform of
x(n) =
A, −M ≤ n ≤ M
0, elsewhere
x(n) is real and even (x(−n) = x(n))
XR(ω) = x(0) + 2
∞
n=1
x(n) cos ωn (even) and XI (ω) = 0
X(ω) = XR(ω) = A 1 + 2
M
n=1
cos ωn = A
sin(M + 1
2)ω
sin(ω/2)
|X(ω)| = A
sin(M + 1
2)ω
sin(ω/2)
and X(ω) =
0, if X(ω) > 0
π, if X(ω) < 0
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Properties of Fourier Transform for Discrete-Time Signals
Figure 6: Spectral characteristics of rectangular pulse in Example.
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Fourier Transform Theorems and Properties
Linearity
If
x1(n)
F
←→ X1(ω)
and
x2(n)
F
←→ X2(ω)
then
a1x1(n) + a2x2(n)
F
←→ a1X1(ω) + a2X2(ω)
Fourier transformation, viewed as an operation on a signal x(n), is a
linear transformation
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Fourier Transform Theorems and Properties
Example
Determine Fourier transform of
x(n) = a|n|, −1 < a < 1
x(n) can be expressed as
x(n) = x1(n) + x2(n)
where
x1(n) =
an, n ≥ 0
0, n < 0
and x2(n) =
a−n, n < 0
0, n ≥ 0
Fourier transform of x1(n)
X1(ω) =
∞
n=−∞
x1(n)e−jωn
=
∞
n=0
an
e−jωn
=
∞
n=0
(ae−jω
)n
=
1
1 − ae−jω
knowing that |ae−jω| = |a| < 1
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Fourier Transform Theorems and Properties
Example (continued)
Fourier transform of x2(n)
X2(ω) =
∞
n=−∞
x2(n)e−jωn
=
−1
n=−∞
a−n
e−jωn
=
−1
n=−∞
(aejω
)−n
=
∞
k=1
(aejω
)k
=
aejω
1 − aejω
Combining these two transforms, we obtain Fourier transform of x(n)
X(ω) = X1(ω) + X2(ω) =
1 − a2
1 − 2a cos ω + a2
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Fourier Transform Theorems and Properties
Figure 7: Sequence x(n) and its Fourier transform in Example with a = 0.8.
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Fourier Transform Theorems and Properties
Time shifting
If
x(n)
F
←→ X(ω)
then
x(n − k)
F
←→ e−jωk
X(ω)
Proof
F{x(n)} = X(ω) =
∞
n=−∞
x(n)e−jωn
F{x(n − k)} =
∞
n=−∞
x(n − k)e−jωn
l=n−k
−−−−→ =
∞
l=−∞
x(l)e−jω(l+k)
= X(ω)e−jωk
= |X(ω)|ej[ X(ω)−ωk]
If a signal is shifted in time domain by k samples, its magnitude
spectrum remains unchanged, but phase spectrum is changed by an
amount −ωk
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Fourier Transform Theorems and Properties
Time reversal
If
x(n)
F
←→ X(ω)
then
x(−n)
F
←→ X(−ω)
Proof
F{x(n)} = X(ω) =
∞
n=−∞
x(n)e−jωn
F{x(−n)} =
∞
n=−∞
x(−n)e−jωn
l=−n
−−−→ =
∞
l=−∞
x(l)ejωl
= X(−ω) = |X(−ω)|ej X(−ω)
if x(n) is real
−−−−−−−−−−→ = |X(ω)|e−j X(ω)
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Fourier Transform Theorems and Properties
Convolution theorem
If
x1(n)
F
←→ X1(ω)
and
x2(n)
F
←→ X2(ω)
then
x(n) = x1(n) ∗ x2(n)
F
←→ X(ω) = X1(ω)X2(ω)
Proof: multiply both sides of convolution formula by e−jωn
and sum
over all n
x(n) = x1(n) ∗ x2(n) =
∞
k=−∞
x1(k)x2(n − k)
X(ω) =
∞
n=−∞
x(n)e−jωn
=
∞
n=−∞
∞
k=−∞
x1(k)x2(n − k) e−jωn
n−k=l
−−−−→ =
∞
k=−∞
x1(k)
∞
l=−∞
x2(l)e−jω(k+l)
= X1(ω)X2(ω)
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Fourier Transform Theorems and Properties
Example
Using convolution theorem, determine convolution of sequences
x1(n) = x2(n) = {1, 1
↑
, 1}
For real and even signals
XR (ω) = x(0) + 2
∞
n=1
x(n) cos ωn and XI (ω) = 0
X1(ω) = X2(ω) = 1 + 2 cos ω
X(ω) = X1(ω)X2(ω) = (1 + 2 cos ω)2
= 3 + 4 cos ω + 2 cos 2ω
= 3 + 2(ejω
+ e−jω
) + (ej2ω
+ e−j2ω
)
Thus convolution of x1(n) with x2(n) is (recall X(ω) =
∞
n=−∞ x(n)e−jωn
)
x(n) = {1, 2, 3
↑
, 2, 1}
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Fourier Transform Theorems and Properties
Figure 8: Graphical representation of the convolution property.
27 / 40
Fourier Transform Theorems and Properties
Correlation theorem
If
x1(n)
F
←→ X1(ω)
and
x2(n)
F
←→ X2(ω)
then
rx1x2
(n)
F
←→ Sx1x2
(ω) = X1(ω)X2(−ω)
Proof: multiply both sides of correlation formula by e−jωn
and sum over
all n
rx1x2 (n) =
∞
k=−∞
x1(k)x2(k − n)
Sx1x2 (ω) =
∞
n=−∞
rx1x2 (n)e−jωn
=
∞
n=−∞
∞
k=−∞
x1(k)x2(k − n) e−jωn
k−n=l
−−−−→ =
∞
k=−∞
x1(k)
∞
l=−∞
x2(l)e−jω(k−l)
= X1(ω)X2(−ω)
Sx1x2 (ω) is called cross-energy density spectrum of signals x1(n) and x2(n)
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Fourier Transform Theorems and Properties
Wiener-Khintchine theorem
Let x(n) be a real signal. Then
rxx (l)
F
←→ Sxx (ω)
I.e., energy spectral density of an energy signal is Fourier transform of its
autocorrelation sequence
This is a special case of preceding theorem (correlation theorem)
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Fourier Transform Theorems and Properties
Example
Determine energy density spectrum of
x(n) = anu(n), −1 < a < 1
Using results of previous examples for this signal
rxx (l) =
1
1 − a2
a|l|
, −∞ < l < ∞
F{rxx (l)} =
1
1 − a2
F{a|l|
} =
1
1 − 2a cos ω + a2
According to Wiener-Khintchine theorem
Sxx (ω) =
1
1 − 2a cos ω + a2
30 / 40
Fourier Transform Theorems and Properties
Frequency shifting
If
x(n)
F
←→ X(ω)
then
ejω0n
x(n)
F
←→ X(ω − ω0)
Proof
X(ω) =
∞
n=−∞
x(n)e−jωn
−→ X(ω − ω0) =
∞
n=−∞
x(n)e−j(ω−ω0)n
=
∞
n=−∞
(ejω0n
x(n))e−jωn
= F{ejω0n
x(n)}
31 / 40
Fourier Transform Theorems and Properties
Figure 9: Illustration of the frequency-shifting property of the Fourier transform
(ω0 ≤ 2π − ωm).
32 / 40
Fourier Transform Theorems and Properties
Modulation theorem
If
x(n)
F
←→ X(ω)
then
x(n) cos ω0n
F
←→ 1
2 [X(ω + ω0) + X(ω − ω0)]
Proof: expressing cos ω0n as
cos ω0n = 1
2 (ejω0n
+ e−jω0n
)
and using frequency-shifting property
x(n)
F
←→ X(ω) −→ ejω0n
x(n)
F
←→ X(ω − ω0)
we obtain
F{1
2 (ejω0n
+ e−jω0n
)x(n)} = 1
2 [X(ω − ω0) + X(ω + ω0)]
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Fourier Transform Theorems and Properties
Figure 10: Graphical representation of the modulation theorem; the spectra of
the signals x(n), y1(n) = x(n) cos 0.5πn and y2(n) = x(n) cos πn.
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Fourier Transform Theorems and Properties
Parseval’s theorem
If
x1(n)
F
←→ X1(ω)
and
x2(n)
F
←→ X2(ω)
then
∞
n=−∞
x1(n)x∗
2 (n) =
1
2π
π
−π
X1(ω)X∗
2 (ω)dω
Proof: eliminating X1(ω) on right-hand side of above equation
1
2π 2π
∞
n=−∞
x1(n)e−jωn
X∗
2 (ω)dω
=
∞
n=−∞
x1(n)
1
2π 2π
X∗
2 (ω)e−jωn
dω =
∞
n=−∞
x1(n)x∗
2 (n)
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Fourier Transform Theorems and Properties
In special case where x2(n) = x1(n) = x(n), Parseval’s relation reduces
to
∞
n=−∞
|x(n)|2
=
1
2π 2π
|X(ω)|2
dω
Left-hand side of this equation is energy Ex of x(n)
Left-hand side is also equal to autocorrelation of x(n), rxx (l) at l = 0
Integrand in right-hand side is equal to energy density spectrum, so
integral over −π ≤ ω ≤ π yields total signal energy
Ex = rxx (0) =
∞
n=−∞
|x(n)|2
=
1
2π 2π
|X(ω)|2
dω =
1
2π
π
−π
Sxx (ω)dω
36 / 40
Fourier Transform Theorems and Properties
Multiplication of two sequences (Windowing theorem)
If
x1(n)
F
←→ X1(ω)
and
x2(n)
F
←→ X2(ω)
then
x3(n) = x1(n)x2(n)
F
←→ X3(ω) =
1
2π
π
−π
X1(λ)X2(ω − λ)dλ
Integral on right-hand side is convolution of X1(ω) and X2(ω)
This convolution integral is known as periodic convolution of X1(ω) and
X2(ω) because it is convolution of two periodic functions having the
same period
Multiplication of aperiodic sequences is equivalent to periodic
convolution of their Fourier transforms
Based on duality, convolution in time domain (aperiodic summation) is
equivalent to multiplication of continuous periodic Fourier transforms
Due to periodicity of Fourier transforms for discrete-time signals, there is
no ”perfect” duality between time and frequency domains with respect
to convolution operation, as in the case of continuous-time signals
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Fourier Transform Theorems and Properties
Proof of windowing theorem:
We know
x3(n) = x1(n)x2(n) and x1(n) =
1
2π
π
−π
X1(λ)ejλn
dλ
Thus
X3(ω) =
∞
n=−∞
x3(n)e−jωn
=
∞
n=−∞
x1(n)x2(n)e−jωn
=
∞
n=−∞
1
2π
π
−π
X1(λ)ejλn
dλ x2(n)e−jωn
=
1
2π
π
−π
X1(λ)dλ
∞
n=−∞
x2(n)e−j(ω−λ)n
=
1
2π
π
−π
X1(λ)X2(ω − λ)dλ
38 / 40
Fourier Transform Theorems and Properties
Differentiation in frequency domain
If
x(n)
F
←→ X(ω)
then
nx(n)
F
←→ j dX(ω)
dω
Proof: differentiate series in Fourier transform definition, term by term
with respect to ω
X(ω) =
∞
n=−∞
x(n)e−jωn
dX(ω)
dω
=
d
dω
∞
n=−∞
x(n)e−jωn
=
∞
n=−∞
x(n)
d
dω
e−jωn
= −j
∞
n=−∞
nx(n)e−jωn
Multiplying both sides by j, we obtain the desired result 39 / 40
References
John G. Proakis, Dimitris G. Manolakis, Digital Signal
Processing: Principles, Algorithms, and Applications, Prentice
Hall, 2006.
40 / 40