Chapter 5 Frequency Domain Analysis of Systems

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Chapter 5Frequency Domain Analysis of Systems
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CT, LTI Systems

• Consider the following CT LTI system
• Assumption the impulse response h(t) is
  absolutely integrable, i.e.,

(this has to do with system stability (ECE 352))
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Response of a CT, LTI System to a Sinusoidal
Input

• Whats the response y(t) of this system to the
  input signal
• We start by looking for the response yc(t) of the
  same system to

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Response of a CT, LTI System to a Complex
Exponential Input

• The output is obtained through convolution as

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The Frequency Response of a CT, LTI System

• By defining
• it is
• Therefore, the response of the LTI system to a
  complex exponential is another complex
  exponential with the same frequency

is the frequency response of the CT,
LTI system Fourier transform of h(t)
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Analyzing the Output Signal yc(t)

• Since is in general a complex
  quantity, we can write

output signals phase
output signals magnitude
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Response of a CT, LTI System to a Sinusoidal
Input

• With Eulers formulas we can express
• as
• and, by exploiting linearity, it is

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Response of a CT, LTI System to a Sinusoidal
Input Contd

• Thus, the response to
• is
• which is also a sinusoid with the same
  frequency but with the amplitude scaled by
  the factor and with the phase
  shifted by amount

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DT, LTI Systems

• Consider the following DT, LTI system
• The I/O relation is given by

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Response of a DT, LTI System to a Complex
Exponential Input

• If the input signal is
• Then the output signal is given by
• where

is the frequency response of the DT,
LTI system DT Fourier transform (DTFT) of hn
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Response of a DT, LTI System to a Sinusoidal Input

• If the input signal is
• Then the output signal is given by

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Example Response of a CT, LTI System to
Sinusoidal Inputs

• Suppose that the frequency response of a CT, LTI
  system is defined by the following specs

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Example Response of a CT, LTI System to
Sinusoidal Inputs Contd

• If the input to the system is
• Then the output is

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Example Frequency Analysis of an RC Circuit

• Consider the RC circuit shown in figure

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Example Frequency Analysis of an RC Circuit
Contd

• From ENGR 203, we know that
• The complex impedance of the capacitor is equal
  to where
• If the input voltage is , then
  the output signal is given by

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Example Frequency Analysis of an RC Circuit
Contd

• Setting , it is
• whence we can write
• where

and
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Example Frequency Analysis of an RC Circuit
Contd
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Example Frequency Analysis of an RC Circuit
Contd

• The knowledge of the frequency response
  allows us to compute the response y(t) of the
  system to any sinusoidal input signal
• since

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Example Frequency Analysis of an RC Circuit
Contd

• Suppose that and that
• Then, the output signal is

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Example Frequency Analysis of an RC Circuit
Contd
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Example Frequency Analysis of an RC Circuit
Contd

• Suppose now that

• Then, the output signal is

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Example Frequency Analysis of an RC Circuit
Contd
The RC circuit behaves as a lowpass filter, by
letting low-frequency sinusoidal signals pass
with little attenuation and by significantly
attenuating high-frequency sinusoidal signals
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Response of a CT, LTI System to Periodic Inputs

• Suppose that the input to the CT, LTI system is
  a periodic signal x(t) having period T
• This signal can be represented through its
  Fourier series as

where
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Response of a CT, LTI System to Periodic Inputs
Contd

• By exploiting the previous results and the
  linearity of the system, the output of the system
  is

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Example Response of an RC Circuit to a
Rectangular Pulse Train

• Consider the RC circuit
• with input

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Example Response of an RC Circuit to a
Rectangular Pulse Train Contd

• We have found its Fourier series to be
• with

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Example Response of an RC Circuit to a
Rectangular Pulse Train Contd

• Magnitude spectrum of input signal x(t)

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Example Response of an RC Circuit to a
Rectangular Pulse Train Contd

• The frequency response of the RC circuit was
  found to be
• Thus, the Fourier series of the output signal is
  given by

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Example Response of an RC Circuit to a
Rectangular Pulse Train Contd
filter more selective
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Example Response of an RC Circuit to a
Rectangular Pulse Train Contd
filter more selective
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Example Response of an RC Circuit to a
Rectangular Pulse Train Contd
filter more selective
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Response of a CT, LTI System to Aperiodic Inputs

• Consider the following CT, LTI system
• Its I/O relation is given by
• which, in the frequency domain, becomes

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Response of a CT, LTI System to Aperiodic Inputs
Contd

• From , the
  magnitude spectrum of the output signal y(t) is
  given by
• and its phase spectrum is given by

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Example Response of an RC Circuit to a
Rectangular Pulse

• Consider the RC circuit
• with input

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Example Response of an RC Circuit to a
Rectangular Pulse Contd

• The Fourier transform of x(t) is

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Example Response of an RC Circuit to a
Rectangular Pulse Contd
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Example Response of an RC Circuit to a
Rectangular Pulse Contd
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Example Response of an RC Circuit to a
Rectangular Pulse Contd
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Example Response of an RC Circuit to a
Rectangular Pulse Contd

• The response of the system in the time domain can
  be found by computing the convolution
• where

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Example Response of an RC Circuit to a
Rectangular Pulse Contd
filter more selective
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Example Attenuation of High-Frequency Components
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Example Attenuation of High-Frequency Components
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Filtering Signals

• The response of a CT, LTI system with frequency
  response to a sinusoidal signal
• Filtering if or
• then or

is
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Four Basic Types of Filters
lowpass
highpass
passband
stopband
stopband
cutoff frequency
bandpass
bandstop
(many more details about filter design in ECE
464/564 and ECE 567)
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Phase Function

• Filters are usually designed based on
  specifications on the magnitude response
• The phase response has to be taken
  into account too in order to prevent signal
  distortion as the signal goes through the system
• If the filter has linear phase in its
  passband(s), then there is no distortion

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Linear-Phase Filters

• A filter is said to have linear phase
  if
• If is in passband of a linear phase
  filter, its response to
• is

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Ideal Linear-Phase Lowpass

• The frequency response of an ideal lowpass filter
  is defined by

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Ideal Linear-Phase Lowpass Contd

• can be written as
• whose inverse Fourier transform is

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Ideal Linear-Phase Lowpass Contd
Notice the filter is noncausal since is
not zero for
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Ideal Sampling

• Consider the ideal sampler
• It is convenient to express the sampled signal
  as where

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Ideal Sampling Contd

• Thus, the sampled waveform is
• is an impulse train whose weights
  (areas) are the sample values of the
  original signal x(t)

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Ideal Sampling Contd

• Since p(t) is periodic with period T, it can be
  represented by its Fourier series

sampling frequency (rad/sec)
where
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Ideal Sampling Contd

• Therefore
• and
• whose Fourier transform is

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Ideal Sampling Contd
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Signal Reconstruction

• Suppose that the signal x(t) is bandlimited with
  bandwidth B, i.e.,
• Then, if the replicas of
  in
• do not overlap and can be recovered
  by applying an ideal lowpass filter to
  (interpolation filter)

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Interpolation Filter for Signal Reconstruction
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Interpolation Formula

• The impulse response h(t) of the interpolation
  filter is
• and the output y(t) of the interpolation
  filter is given by

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Interpolation Formula Contd

• But
• whence
• Moreover,

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Shannons Sampling Theorem

• A CT bandlimited signal x(t) with frequencies no
  higher than B can be reconstructed from its
  samples if the samples are
  taken at a rate
• The reconstruction of x(t) from its samples
  is provided by the
  interpolation formula

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Nyquist Rate

• The minimum sampling rate is called the
  Nyquist rate
• Question Why do CDs adopt a sampling rate of
  44.1 kHz?
• Answer Since the highest frequency perceived by
  humans is about 20 kHz, 44.1 kHz is slightly more
  than twice this upper bound

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Aliasing
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Aliasing Contd

• Because of aliasing, it is not possible to
  reconstruct x(t) exactly by lowpass filtering the
  sampled signal
• Aliasing results in a distorted version of the
  original signal x(t)
• It can be eliminated (theoretically) by lowpass
  filtering x(t) before sampling it so that
  for