Outline
• Response due to sinusoidal input.
• Frequency response of DT systems.
• Properties of the frequency response of
DT Systems .
• MATLAB commands.
• Sampling theorem and the selection of the
sampling period.
Frequency Response
of Discrete-Time Systems
M. Sami Fadali
Professor of Electrical Engineering
UNR
1
2
Impulse Sampled Representation
of DT Waveform
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.
3
∗
• Laplace transform
∗
∗
∗
∗
• Define transfer function for sampled inputs
4
Verification Without
Impulse Sampling
Frequency Response
∗
∗
• Substitute
∗
• z-transform
• Frequency response
∗
Example:
• System z-transfer function
∗
∗
5
Response due to Sinusoid
6
Partial Fraction Coefficient
• System Output:
• Poles inside the unit circle  for sufficiently large
• Partial fraction coefficient
• Partial Fraction Expansion (divide by )
• Inverse z-transform
7
8
Steady-state Output
Response to Sampled Sinusoid
∠
• Real part = response due to a sampled cosine input
• Imaginary part = response for a sampled sine input
• Sampled cosine response
• Similar sampled sine response with sine replacing
cosine.
• Sinusoid of the same frequency scaled and
phase shifted (resp.) by the magnitude and
angle
• Frequency response function obtained
earlier using impulse sampling.
• Use complex arithmetic to determine the
steady-state response due to a sampled
sinusoid without z-transformation.
9
Example
Properties of the Frequency
Response of DT Systems
Find the steady-state response of the system
1. DC Gain.
2. Periodic nature of
frequency response.
3. Symmetry.
due to the sampled sinusoid
Solution: For large with
.
3
.
cos 0.2
1
.
0.1
∠
cos 0.2
∠
6.4 cos 0.2
10
.
0.5
1
.
0.1
.
0.5
0.614
11
12
DC Gain
Periodic Nature
1. DC Gain: The DC gain is equal to
Frequency response is a periodic function of
rad/s.
frequency with period
Proof:
Proof
→
→
• Complex exponential: periodic with period
rad/s.
single-valued function of its argument.
• It also is periodic and has the same repetition
frequency.
13
14
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
15
a)We only need
for frequencies
from DC to
.
b)Obtain frequency response for
by
symmetry.
is
c)Frequency response
periodically repeated for
.
d)Negligible frequency response amplitudes
for
 no overlap of
repeated frequency response cycles.
16
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.
17
Spectrum of Sampled Waveform
18
MATLAB Commands
Calculate & plot frequency response of DT system
• Spectrum of sampled waveform
= periodic function of with period
is a real valued function
1) Magnitude = even function of frequency
2) Phase = odd function of frequency
Sampling period T = 0.2 s
>> z=tf('z',0.2)% Define operator z
>> g= 0.1*(z–0.01)/(z–0.05) % Transfer function
>> z1= 0.1+j*0.1
>> f_resp = evalfr(g,z1) % Evaluate at z=z1
>> H = freqresp(g,w) % Evaluate at freq. grid w
19
20
Frequency Response Plots
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(:) )
Several Plots on One Page
>> subplot(2, 3, 4) % rows, columns, order
i) creates a 2-row, 3-column grid
ii) draw axes at the first position of the second row
(the first three plots are in the first row)
Next, plot command uses axes
>> bode(g) % Bode plot in location (2,3,4)
21
22
The Sampling Theorem
The Sampling Theorem
Theorem 2.4 The band-limited signal
F
can be reconstructed from the discrete-time
waveform
∗
Two different waveforms with
identical samples.
if and only if
• Use an ideal low pass filter of bandwidth
23
24
Proof
Ideal LPF
• Unit impulse train and its Fourier transform
F
For a band-limited signal, the amplitude
and phase in the frequency range 0 to
can be recovered by an ideal low-pass
filter.
• Impulse sampling: multiply waveform
by
• Spectrum of product: convolution of two spectra.
F
25
26
Finite Bandwidth Approximation
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.
27







Time-limiting: pulse  function of infinite duration .
Frequency Convolution: time multiplication is
equivalent to convolution of Fourier transforms.
Spectrum of time-limited function = convolution of its
spectrum with a sinc function (infinite BW).
Spectrum of a time limited function: infinite BW.
Measurements over a finite time period: infinite BW.
Treat physical signals as band-limited: negligible
spectral components beyond "effective bandwidth“.
Choose a suitable sampling rate using the sampling
theorem.
28
Choice of Sampling Rate
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.
• Lower bound specified in sampling
theorem.
• Rule of thumb: choose
• Constant
depends on the application.
29
Linear System
30
Second Order System
Output spectrum=frequency response  input spectrum
• Input is not known a priori  base our choice of
sampling frequency on the frequency response.
• First order system
• K = DC gain,
• BW of the system is approximated by
• Choose sampling frequency
= system bandwidth
• Choose sampling frequency (assume
)
• Step response of a second order system includes
31
32
Example 2.24
Example 2.23
• Design a closed-loop control system
• Design Specs.:
For a signal of bandwidth 10 rad/s, select a
suitable sampling frequency and find the
corresponding sampling period.
Solution:
• Choose sampling frequency
rad/s.
• The corresponding sampling period
– 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.
33
Solution
Sampling period
(a)
Sensor delay=0.005 s
Choose
(b)
s.
s > sensor delay.
Sensor delay=0.02 s
Choose
s = sensor delay.
35
34