MAE143A Signals & Systems Winter 2016
Discrete-time signals
Generating discrete-time signals by sampling continuoustime signals will be a major subject of this course
Some signals are inherently discrete-time, e.g. sunset time
We consider periodic sampling
Fixed time between samples: Ts seconds
Ts is the sampling period
1/Ts Hz is the sampling rate or sampling frequency
The interpretation of a discrete-time signal relies on
knowing the sampling rate
There other sampling strategies such as event-based or
event-triggered sampling. We do not study these.
1.5 Discrete Signals
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2
MAE143A Signals & Systems Winter 2016
Sampled exponential x(t) = exp(1.5 t)
Continuous-time exponential: samping at 100Hz, 20Hz, 30 Hz
100Hz
contiuous
100Hz
20Hz
30Hz
5
20Hz
30Hz
signal (units)
4
3
2
1
0
0.88
1.5 Discrete Signals
0.89
0.9
0.91
0.92
0.93
time (s)
0.94
0.95
0.96
0.97
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MAE143A Signals & Systems Winter 2016
Sampled exponential continued
Sampling period T (0.01s, 0.05s, 1/30s)
Continuous signal (function) x(t)
x[n] = x(nT )
Discrete signal (vector) x[n]
Continuous-time exponential: samping at 100Hz, 20Hz, 30 Hz
contiuous
100Hz
20Hz
30Hz
5
4
signal (units)
In electronics, this is done by
a circuit
-  sample and hold and
-  analog to digital converter
3
2
1
0
0.88
1.5 Discrete Signals
0.89
0.9
0.91
0.92
0.93
time (s)
0.94
0.95
0.96
0.97
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MAE143A Signals & Systems Winter 2016
Sampled signals … for now x[n] = x(nT )
We shall study sampling in greater detail later
It is a nuanced field … and very important!
2
3
1.0779
61.16187
6
7
0
7
1.2523
T = 0.05s, n = [1 : 5] , x[n] = 6
6
7
41.34995
1.4550
x(t) = exp(1.5 t)
A sampled signal is an ordered sequence, which is the
same as a vector
We can consider sequences with a (countably) infinite
number of elements
1.5 Discrete Signals
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MAE143A Signals & Systems Winter 2016
Discrete impulse
1,
0,
[n] =
n=0
else
Discrete impulse function
1.2
1
0.8
signal (units)
Discrete impulse and step
signals
(
0.6
0.4
0.2
Not the sampled version of
(t)
0
-0.2
-3
-2
-1
0
1
2
3
4
5
time (samples)
1[n] =
No craziness of functions
Note:
1[n] =
n
X
k= 1
1.5 Discrete Signals
1,
0,
n 0
else
Discrete step function
1.2
1
0.8
signal (units)
Discrete step
(
0.6
0.4
0.2
[k]
0
-0.2
-3
-2
-1
0
1
time (samples)
2
3
4
5
MAE143A Signals & Systems Winter 2016
Sampling sinusoids
Jump to Matlab for demo
[diary file on class website: jan14.txt]
1.5 Discrete Signals
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7
MAE143A Signals & Systems Winter 2016
The Z-transform of a discrete signal
Discrete signal x[n], n = . . . ,
Z-transform of the signal
2, 1, 0, 1, 2, . . .
X (z) =
1
X
z
k
x[k]
k=0
Discrete-time alternate to the Laplace transform for
continuous-time signals
z is a complex variable in the complex z-plane
Just like the Laplace transform, there are unilateral k 2 [0, 1)
and bilateral versions k 2 ( 1, 1)
We will be concerned only with the unilateral version
1.5 Discrete Signals
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MAE143A Signals & Systems Winter 2016
Computing Z transforms
Sampled exponential example, sample rate 20Hz
continuous-time signal x(t) = e1.5t
1
sample period Ts = 20 s = 0.05 s
sample times for n = {0, 1, 2, 3, 4, . . . }
nTs = {0, 0.05, 0.10, 0.15, 0.20, . . . } s
sample values x[n] = {e0 , e1.5⇥0.05 , e1.5⇥0.10 , e1.5⇥0.15 , . . . }
x[k] =
Z transform
X (z) =
=
1.5 Discrete Signals
1
X
k=0
1
z
(
k
e1.5⇥0.05⇥k ,
0,
x[k] =
1
X
z
k 0,
else.
k 1.5⇥0.05⇥k
e
k=0
1
=
1
1.5⇥0.05
z e
z
=
1
X
k=0
z
e1.5⇥0.05
z
1 1.5⇥0.05 k
e
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MAE143A Signals & Systems Winter 2016
Computing Z transforms 2
Geometric series formula for any value of a
a)(1 + a + a2 + a3 + · · · + aN ) = 1
(1
2
3
1 + a + a + a + ··· + a
Infinite sums
1
X
z
1 1.5⇥0.05
e
1.5 Discrete Signals
1
k
a =
Our Z transformX
1
provided
k
z
1
a
k=0
<1
or
1
aN +1
1 a
1
aN +1
1 a
provided |a| < 1
1 1.5⇥0.05 k
e
=
a =
k=0
k=0
X (z) =
N
X
N
aN +1
=
1
1
=
z 1 e1.5⇥0.05
z
|z| > e1.5⇥0.05
z
e1.5⇥0.05
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MAE143A Signals & Systems Winter 2016
Computing Z transforms 3
Higher-order poles
(1
1
=
z 1 a)2
✓
1
z
1
1
= (1 + z
=1+z
1a
1
◆✓
a+z
1
1a
2 2
3 3
a +z
2
2a + z
1
z
◆
a + . . . )(1 + z
3a2 + z
3
1
2 2
a+z
a +z
3 3
a + ...)
4a3 + . . .
= Z {(n + 1)an }
(1
1
=
z 1 a)3
✓
1
1
z
= (1 + z
1a
1
◆✓
a+z
(1
1
z 1 a)2
2 2
a +z
= 1 + z 1 3a + z 2 6a2 + z
⇢
n+2
=Z
(n + 1)an
2
1.5 Discrete Signals
3 3
◆
a + . . . )(1 + z
3
10a3 + . . .
1
2a + z
2
3a2 + z
3
4a3 + . . . )
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MAE143A Signals & Systems Winter 2016
Z transforms of sampled exponential signals
Consider a continuous-time (complex) exponential signal
x(t) = ebt with b a complex number
nbT
Sample this at sampling period Ts x[n] = e s
Take the Z transform of this discrete-time signal
1
X
X (z) =
z
k kbTs
e
k=0
=
1
X
z
1 bTs k
e
k=0
=
1
1
1 ebTs
z
=
z
z
ebTs
Convergence provided |z| > ebTs = eRe(b)Ts
L e
bt
=
1
s
b
pole at s=b
convergence if Re(s)>Re(b)
1.5 Discrete Signals
Z e
nbt
=
z
z
ebTs
pole at z=ebTs
convergence if
|z| > ebTs |z| > eRe(b)Ts
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MAE143A Signals & Systems Winter 2016
Foreshadowing MAE143C Digital Control
L e
bt
=
1
s
Z e
b
nbt
pole at s=b
convergence if Re(s)>Re(b)
Ts +j!Ts
z
pole at z=ebTs
convergence if |z| > ebTs
General transformation between s and z:
z = esTs = e
=
z
ebTs
=e
z = esTs
Ts j!Ts
e
So
Re(s) > Re(b) , |z| > ebTs
Also
Re(s) < 0 , |z| < 1
The open left half s-plane corresponds to the inside of the unit disk
in the z-plane
1.5 Discrete Signals
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MAE143A Signals & Systems Winter 2016
Summary
Continuous-time signals
Discrete-time signals
t takes values in a real
interval
Signal x(t) is a real
function
Laplace transforms are
used L {f (t)} = F (s)
s is a complex variable
Fourier series for periodic
signals
Fourier transform for
bounded energy signals
n takes integer values
1.5 Discrete Signals
Signal x[n] is a real
sequence
Z transforms are used
Z {x[n]} = X(z)
z is a complex variable
Discrete (Fast) Fourier
Transform used to
analyze a finite sequence