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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
CHAPTER 6Frequency Response, Bode
Plots, and Resonance1. State the fundamental concepts of Fourier
analysis.
2. Determine the output of a filter for a given input consisting of sinusoidal components using the filters transfer function.
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
3. Use circuit analysis to determine the transfer functions of simple circuits.
4. Draw first-order lowpass or highpass filter circuits and sketch their transfer functions.
5. Understand decibels, logarithmic frequency scales, and Bode plots.
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
6. Draw the Bode plots for transfer functions of first-order filters.
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
Fourier Analysis
All real-world signals are sums of sinusoidal components having various frequencies, amplitudes, and phases.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
Filters
Filters process the sinusoid components of an inputsignal differently depending of the frequency of each component. Often, the goal of the filter is to retain the components in certain frequency ranges and to reject components in other ranges.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
Transfer Functions
The transfer function H(f ) of the two-port filter is defined to be the ratio of the phasor output voltage to the phasor input voltage as a function of frequency:
( )in
out
VV=fH
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
The magnitude of the transfer function shows how the amplitude of each frequency component is affected by the filter. Similarly, the phase of the transfer function shows how the phase of eachfrequency component is affected by the filter.
( )in
out
VV=fH
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
Determining the output of a filter for an input with multiple
components:
1. Determine the frequency and phasor representation for each input component.
2. Determine the (complex) value of the transfer function for each component.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
3. Obtain the phasor for each output component by multiplying the phasor for each input component by the corresponding transfer-function value.
4. Convert the phasors for the output components into time functions of various frequencies. Add these time functions to produce the output.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
Linear circuits behave as if they:
1. Separate the input signal into components having various frequencies.
2. Alter the amplitude and phase of each component depending on its frequency.
3. Add the altered components to produce the output signal.
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
FIRST-ORDER LOWPASSFILTERS
RCfB 2
1=
( ) ( )BffjfH
+=
11
( )( )211
BfffH
+=
( )
=
BfffH arctan
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
( )( )211
BfffH
+=
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
DECIBELS, THE CASCADE CONNECTION, AND LOGARITHMIC
FREQUENCY SCALES
( ) ( )fHfH log20dB
=
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
Cascaded Two-Port Networks
( ) ( ) ( )fHfHfH 21 =
( ) ( ) ( )dB2dB1dB
fHfHfH +=
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
Logarithmic Frequency Scales
On a logarithmic scale, the variable is multiplied by a given factor for equalincrements of length along the axis.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
A decade is a range of frequencies for which the ratio of the highest frequency to the lowest is 10.
=
1
2log decades ofnumber ff
An octave is a two-to-one change in frequency.
( )( )
=
=
2logloglog octaves ofnumber 12
1
22
ffff
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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Chapter 6Frequency Response, Bode Plots, and Resonance
BODE PLOTSA Bode plot shows the magnitude of a network function in decibels versus frequency using a logarithmic scale for frequency.
( ) ( )BffjfH
+=
11
( )
+=
2
dB1log10
BfffH
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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Chapter 6Frequency Response, Bode Plots, and Resonance
1. A horizontal line at zero for f < fB /10.
2. A sloping line from zero phase at fB /10 to 90 at 10fB.
3. A horizontal line at 90 for f > 10fB.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
FIRST-ORDER HIGHPASS FILTERS
( ) ( )( )BB
ffjffjfH
+==
1inout
VV
RCfB 2
1=
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
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Chapter 6Frequency Response, Bode Plots, and Resonance
SERIES RESONANCE
Resonance is a phenomenon that can be observed in mechanical systems andelectrical circuits.
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ELECTRICAL
ENGINEERINGPrinciples and
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SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
LCf
21
0 =
RLfQs 0
2=
CRfQs
021
=
( )
+=
ff
ffjQRfZ ss 00
1
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
Series Resonant Circuit as a Bandpass Filter
( )ffffjQssR
0011
+=
VV
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
LH ffB =
sQfB 0=
20BffH +
20BffL
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
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Chapter 6Frequency Response, Bode Plots, and Resonance
PARALLEL RESONANCE
( ) ( )fLjfCjRZ p 21211
+=
LCf
21
0 = LfRQp
02= CRfQp 02=
( )ffffjQRZ
pp
001 +=
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
Ideal Filters
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
Second-Order Lowpass Filter
( ) ( )( )ffffjQffjQfH
s
s
00
0
in
out
1 +==
VV
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
Applications
SE OND EDITION
Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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Chapter 6Frequency Response, Bode Plots, and Resonance
DIGITAL SIGNAL PROCESSING
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
Conversion of Signals from Analog to Digital Form
If a signal contains no components with frequencies higher than fH, the signal can be exactly reconstructed from its samples, providedthat the sampling rate fs is selected to be more than twice fH.
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
Digital Lowpass Filter
( ) ( ) ( ) ( )nxanayny += 11
TTa
+
=1
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance
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ELECTRICAL
ENGINEERINGPrinciples and
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Chapter 6Frequency Response, Bode Plots, and Resonance