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Signals and Systems

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Title: Signals and Systems

1
Signals and Systems
2
Signals As Functions of Time
Review

Continuous-time signals are functions of a real
argument
x(t) where time, t, can take any real value
x(t) may be 0 for a given range of values of t
Discrete-time signals are functions of an
argument that takes values from a discrete set
xk where k ? ...-3,-2,-1,0,1,2,3...
Integer time index, e.g. k, for discrete-time
systems
Values for x may be real or complex

3
Analog vs. Digital Signals
Review

Analog
Continuous in both time and amplitude
Digital
Discrete in both time and amplitude

4
The Many Faces of Signals

A function, e.g. cos(t) or cos(p k), useful in
analysis
A sequence of numbers, e.g. 1,2,3,2,1 or a
sampled triangle function, useful in simulation
A collection of properties, e.g. even, causal,
stable, useful in reasoning about behavior
A piecewise representation, e.g.
A generalized function, e.g. d(t)
What everyday device uses twosinusoids to
transmit a digital code?

5
Telephone Touchtone Signal

Dual-tone multiple frequency (DTMF) signaling
Sum of two sinusoids onefrom low-frequency
groupand high-frequency group
On for 40-60 ms and off forrest of signaling
interval(symbol duration)
100 ms for ATT
80 ms for ITU Q.24 standard
Maximum dialing rate
ATT 10 symbols/s (40 bits/s)
Q.24 12.5 symbols/s (50 bits/s)

ITU is the International Telecommunication Union
6
Unit Impulse
Review

Mathematical idealism foran instantaneous event
Dirac delta as generalizedfunction (a.k.a.
functional)
Selected properties
Unit area
Sifting
provided g(t) is defined at t0
Scaling
Note that

Unit Area
e
-e
t
7
Unit Impulse
Review

By convention, plot Diracdelta as arrow at
origin
Undefined amplitude at origin
Denote area at origin as (area)
Height of arrow is irrelevant
Direction of arrow indicates sign of area
With d(t) 0 for t ? 0,it is tempting to think
f(t) d(t) f(0) d(t)
f(t) d(t-T) f(T) d(t-T)

Simplify unit impulse under integration only
8
Unit Impulse
Review

We can simplify d(t) under integration
Assuming ?(t) is defined at t0
What about?
What about?
By substitution of variables,

Other examples
What about at origin?

Before Impulse
After Impulse
9
Unit Impulse Functional

Relationship between unit impulse and unit step
What happens at the origin for u(t)?
u(0-) 0 and u(0) 1, but u(0) can take any
value
Common values for u(0) are 0, ½, and 1
u(0) ½ is used in impulse invariance filter
design

L. B. Jackson, A correction to impulse
invariance, IEEE Signal Processing Letters, vol.
7, no. 10, Oct. 2000, pp. 273-275.
10
Systems
Review

Systems operate on signals to produce new signals
or new signal representations
Continuous-time examples
y(t) ½ x(t) ½ x(t-1)
y(t) x2(t)
Discrete-time system examples
yn ½ xn ½ xn-1
yn x2n

Squaring function can be used in sinusoidal
demodulation
Average of current input and delayed input is a
simple filter
11
System Properties
Review

Let x(t), x1(t), and x2(t) be inputs to a
continuous-time linear system and let y(t),
y1(t), and y2(t) be their corresponding outputs
A linear system satisfies
Additivity x1(t) x2(t) ? y1(t) y2(t)
Homogeneity a x(t) ? a y(t) for any real/complex
constant a
For a time-invariant system, a shift of input
signal by any real-valued t causes same shift in
output signal, i.e. x(t - t) ? y(t - t) for all t

12
System Properties
Review

Ideal delay by T seconds. Linear?
Scale by a constant (a.k.a. gain block)
Two different ways to express it in a block
diagram
Linear?

13
System Properties

Tapped delay line
Linear? Time-invariant?

Each T represents a delay of T time units
There are M-1 delays
Coefficients (or taps) are a0, a1, aM-1
14
System Properties

Amplitude Modulation (AM)
y(t) A x(t) cos(2p fc t)
fc is the carrierfrequency(frequency ofradio
station)
A is a constant
Linear? Time-invariant?
AM modulation is AM radio if x(t) 1 ka m(t)
where m(t) is message (audio) to be broadcast

y(t)
A
x(t)
cos(2 p fc t)
15
System Properties

Frequency Modulation (FM)
FM radio
fc is the carrier frequency (frequency of radio
station)
A and kf are constants
Linear? Time-invariant?

16
Sampling

Many signals originate as continuous-time
signals, e.g. conventional music or voice.
By sampling a continuous-time signal at isolated,
equally-spaced points in time, we obtain a
sequence of numbers
k ? , -2, -1, 0, 1, 2,
Ts is the sampling period.

17
Generating Discrete-Time Signals

Uniformly sampling a continuous-time signal
Obtain xk x(Ts k) for -? lt k lt ?.
How to choose Ts?
Using a formula
xk k2 5k 3, for k ? 0would give the
samples3, -1, -3, -3, -1, 3, ...
What does the sequence looklike in continuous
time?

18
System Properties

Let xk, x1k, and x2k be inputs to a linear
system and let yk, y1k, and y2k be their
corresponding outputs
A linear system satisfies
Additivity x1k x2k ? y1k y2k
Homogeneity a xk ? a yk for any real/complex
constant a
For a time-invariant system, a shift of input
signal by any integer-valued m causes same shift
in output signal, i.e. xk - m ? yk - m, for
all m

19
System Properties