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ENSC380
Lecture 22
Objectives:
• Signals and Systems Fourier Analysis:
• Power Spectrum
• Log-Magnitude response plots
• Bode plots
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2/14
Power Spectrum
• Power spectrum of a signal refers to the amount of power a signal contains in
a very narrow band around each frequency.
• To find the power spectrum of a signal, x(t), the following system is used. This
is an example of the application of BPFs.
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Noise Removal
• An important application of filters (usually LPF or BPF) is to reduce the noise
contents of a signal.
• Noise is an undesirable signal which inevitably adds to signal when they are
stored, transmitted, filtered, . . . .
• Practical signals usually have a limited bandwidth, whereas noise usually has
unlimited or very large bandwidth.
• The idea is to remove the noise in the frequency range that the desired signal
does not exist.
• The ratio of the desired signal power to the noise power is called the signal to
noise ratio (SNR). The goal of a noise reduction system is to increase SNR as
much as possible.
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Log-Magnitude Plot
• Sometimes instead of plotting the magnitude of a frequency response of a
system versus frequency, we plot the logarithm of the magnitude.
• The reason is that logarithm de-emphasizes large values and emphasizes
small values. This helps see the subtle differences between the magnitude
response of two systems more clearly.
H1 (f ) =
1
1 + j2πf
H2 (f ) =
30
30 − 4π 2 f 2 + j62πf
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Bode Plot
• A more common practice is to plot the logarithm of the magnitude of the
frequency response (H(f ) or H(jω )) against a logarithmic scale of the
frequency. This is called a Bode diagram or a Bode Plot.
• In a Bode plot the magnitude of H(jω) is converted into a logarithmic scale
called decibel (dB). The unit Bel is named after Alexander Graham Bell, and is
defined as the base 10 logarithm of the ratio of two powers. Decibel is a
tenth of a Bel.
• If the input and output signals of a system are x(t) and y(t), with powers Px
and Py respectively, then, the ratio of their powers in Bel and Decibel is:
• On the other hand we know that Px is proportional to |X(f )|2 and Py to
|Y (f )|2 . Thus we can write:
log10 (
Py
)=
Px
• Finally, the conversion of |H(f )| to it’s decibel unit can be written as:
HdB (f ) =
Atousa Hajshirmohammadi, SFU
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Bode Plot (Cont.)
• The bode plots of the previous systems H1 (f ) and H2 (f ):
Note: The phase of H(f ) should also be plotted against the logarithmic scale
of frequency.
• In the following slides, we replace H(f ) with H(jω) for simplicity in the
formulas. H(jω) is the same as H(f ) when 2πf is replaced with ω .
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Bode Plot (Cont.)
• We know that any LTI system is defined by a linear constant-coefficient
differential equation:
• Hence, its transfer function (frequency response) has the form:
• If we find the roots of the numerator and denominator of H(jω) and show them
with zi and pi respectively, we can write:
1−
jω
z1
... 1 −
H(jω) = A jω
jω
1− p
1 − p ... 1 −
1
•
1−
jω
z2
2
jω
zN
jω
pD
zi ’s are called the “zeros” and pi ’s the poles of the transfer function. H(jω) is
equal to zero at ω = zi and goes to infinity for ω = pi .
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Bode Plot (Cont.)
• The above system, can be viewed as a cascade of N + D simpler
sub-systems:
• The transfer function of each subsystem (Hl (jω)) has a magnitude and phase:
|Hl (jω)| and ∠Hl (jω)
• The Bode plot for the over all system (H(jω) or (H(f )) is the sum of the bode
plots for each sub-system (both for the magnitude and phase).
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Component Diagrams
• The bode plot for each sub-system is called a “component diagram”.
subsystems can have different forms:
• Sub-systems with a real (non-zero) • Sub-systems with a real (non-zero)
zero:
pole:
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Component Diagrams (Cont).
• Subsystem with a zero at 0
(Differentiator):
• Subsystem with a pole at 0
(Integrator):
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Component Diagrams (Cont).
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Subsystem with complex zero pair
H(jω) =
jω
1−
z1
jω
1−
z2
where z2 = z1∗
Let
ω0 = |z1 |2
ζ=−
Re(z1 )
ω0
Atousa Hajshirmohammadi, SFU
Component Diagrams (Cont).
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Subsystem with complex pole pair
H(jω) = 1−
jω
p1
1
1−
jω
p2
where p2 = p1∗
Let
ω0 = |p1 |2
ζ=−
Re(p1 )
ω0
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Example
Plot the Bode diagram for the given circuit, with C1 = 1 F, C2 = 2 F, Rs = 4Ω,
R1 = 2Ω, R2 = 3Ω
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Example(Cont.)
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Atousa Hajshirmohammadi, SFU