Frequency Response of Discrete-Time Systems: Outline
Frequency Response of Discrete-Time Systems: Outline
Frequency Response of Discrete-Time Systems: Outline
Frequency Response
of Discrete-Time Systems
M. Sami Fadali
Professor of Electrical Engineering
UNR
Sinusoidal SS Response
1. SS Response of an LTI DT system to a
sampled sinusoidal input: sinusoid of
the same frequency as the input with
frequency dependent phase shift and
magnitude scaling.
2. Scale factor and phase shift define a
complex function of frequency: the
frequency response.
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Laplace transform
Verification Without
Impulse Sampling
Frequency Response
Substitute
z-transform
Frequency response
Example:
System Output:
Inverse z-transform
Steady-state Output
Example
1. DC Gain.
2. Periodic nature of
frequency response.
3. Symmetry.
cos 0.2
1
.
0.1
cos 0.2
10
0.5
1
.
0.1
0.5
0.614
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DC Gain
Periodic Nature
Proof
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Observations
Symmetry
For transfer functions with real coefficients
1. Magnitude of TF is an even function of frequency.
2. Phase of TF is an odd function of frequency.
Proof: For negative frequencies, the transfer
function is
For real coefficients
Combine the last two equations
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Observations
e) Sampling with no overlap periodic repetition
of the frequency response of a continuous time
system.
f) Frequency responses of physical systems are
not bandlimited overlapping of the repeated
frequency response cycles (folding).
= the folding frequency.
g)
h) Folding results in distortion of the frequency
response and should be minimized by proper
choice of the sampling frequency
or
filtering.
Frequency Response
of a Digital System.
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MATLAB Commands
Calculate & plot frequency response of DT system
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MATLAB Plots
>> bode( g)
>> nichols( g)
w = frequency grid, use w not wT as in evalfr
>> [Magnitude, Phase, w] = bode(g)
% Bode data
>> [Real, Imag] = nyquist(g, w)
% Nyquist data Output: multidimensional array
>> mag=reshape(Magnitude,1, length(w));
>> plot(w, mag)
>>plot(w, Magnitude(:) )
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if and only if
Use an ideal low pass filter of bandwidth
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Proof
Ideal LPF
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Finite Bandwidth
Idealization associated with infinite duration.
Finite duration implies infinite bandwidth.
Why?
Band limiting: equivalent to multiplication by a
pulse in the frequency domain.
Convolution Theorem: multiplication in the
frequency domain convolution of the inverse
Fourier transforms.
Inverse transform of a band-limited function =
convolution of the original time function with the
sinc function, a function of infinite duration.
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Limitations
1. Sampling frequency upper bounded = sensor
delay.
Example: oxygen sensors used in automotive
air/fuel ratio control have a sensor delay of about 20
ms.
2. Computational time needed to update the control
(less restrictive with the availability of faster
microprocessors).
3. Sampling fast enough to provide a good
representation of the analog physical variables.
Constant
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Linear System
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K = DC gain,
= system bandwidth
)
Step response of a second order system includes
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Example 2.24
Example 2.23
Steady-state error
Damping ratio
Undamped natural frequency
Select a suitable sampling period for the system
if the system has a sensor delay of
(a) 0.005 s
(b) 0.02 s.
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Solution
Sampling period
(a)
Sensor delay=0.005 s
Choose
(b)
s.
s > sensor delay.
Sensor delay=0.02 s
Choose
s = sensor delay.
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