ENSC380Lecture 22

Objectives:

• Signals and Systems Fourier Analysis:• Power Spectrum• Log-Magnitude response plots• Bode plots

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Power Spectrum

• Power spectrum of a signal refers to the amount of power a signal contains ina very narrow band around each frequency.

• To find the power spectrum of a signal, x(t), the following system is used. Thisis 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 noisecontents of a signal.

• Noise is an undesirable signal which inevitably adds to signal when they arestored, transmitted, filtered, . . . .

• Practical signals usually have a limited bandwidth, whereas noise usually hasunlimited or very large bandwidth.

• The idea is to remove the noise in the frequency range that the desired signaldoes not exist.

• The ratio of the desired signal power to the noise power is called the signal tonoise ratio (SNR). The goal of a noise reduction system is to increase SNR asmuch as possible.

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Log-Magnitude Plot

• Sometimes instead of plotting the magnitude of a frequency response of asystem versus frequency, we plot the logarithm of the magnitude.

• The reason is that logarithm de-emphasizes large values and emphasizessmall values. This helps see the subtle differences between the magnituderesponse of two systems more clearly.

H1(f) =1

1 + j2πfH2(f) =

30

30 − 4π2f2 + j62πf

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Bode Plot

• A more common practice is to plot the logarithm of the magnitude of thefrequency response (H(f) or H(jω)) against a logarithmic scale of thefrequency. This is called a Bode diagram or a Bode Plot.

• In a Bode plot the magnitude of H(jω) is converted into a logarithmic scalecalled decibel (dB). The unit Bel is named after Alexander Graham Bell, and isdefined as the base 10 logarithm of the ratio of two powers. Decibel is atenth 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) =

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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 scaleof frequency.

• In the following slides, we replace H(f) with H(jω) for simplicity in theformulas. 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-coefficientdifferential 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 themwith zi and pi respectively, we can write:

H(jω) = A

(

1 − jω

z1

) (

1 − jω

z2

)

. . .(

1 − jω

zN

)

(

1 − jω

p1

) (

1 − jω

p2

)

. . .(

1 − jω

pD

)

• zi’s are called the “zeros” and pi’s the poles of the transfer function. H(jω) isequal 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 simplersub-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 bodeplots 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)zero:

• Sub-systems with a real (non-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).

Subsystem with complex zero pair

H(jω) =

(

1 −jω

z1

) (

1 −jω

z2

)

where z2 = z1∗

Let

ω0 = |z1|2 ζ = −

Re(z1)

ω0

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Component Diagrams (Cont).

Subsystem with complex pole pair

H(jω) =1

(

1 − jω

p1

) (

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.) 14/14

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