Lecture 2
Useful Signal Models
Unit Step Function u(t)
• Step function defined
• Exponential function defined
Useful to describe a signal that begins
at t = 0 (i.e. causal signal).
For example, the signal e-at represents
an everlasting exponential that starts at t = -∞.
The causal for of this exponential can be described as e-at u(t).
Pulse signal
• A pulse signal can be represented by two step
functions
X(t) = u(t-2) – u(t-4)
Example
X(t) = u(t-1)+u(t-2)–4 u(t-3)+u(t-4)+u(t-5)
Ramp signal
The unit ramp function, usually represented as
r(t) , is given by, r(t)=t u(t) or
Example
Example
X(t) = u(t-1)+r(t-1)–r(t-2)-4 u(t-2)+u(t-3)+r(t-3)-r(t-4)
Unit impulse function δ(t)
• First defined by Dirac
Unit impulse
Approximation of an Impulse
Unit impulse function δ(t)
• Other possible approximations to a unit
impulse
• The area under the pulse function must be
unity
• Exponential
Triangular
Gaussian
Multiplying a function Φ(t) by an
impulse
• Since impulse is non-zero only at t = 0, and Φ (t)
at t = 0 is Φ (0), we get:
We can generalize this for t = T:
Sampling Property of Unit Impulse
Function
• Since we have
• It follows that
• This is the same as “sampling” Φ(t) at t = 0.
• If we want to sample Φ (t) at t = T, we just multiple Φ(t) with
• This is called the Sampling or
sifting property of the impulse
The Exponential Function est (1)
• This exponential function is very important in signals
& systems, and the parameter s is a complex variable
given by:
The Exponential Function est (2)
• If σ = 0, then we have the function ejωt, which
has a real frequency of ω
• Therefore the complex variable s =σ + jω is the
complex frequency
• The function est can be used to describe a very
large class of signals and functions. Here are a
number of example:
•
•
•
•
A constant k = k e0t (s = 0)
A monotonic exponential eσt (ω = 0, s = σ)
A sinusoid cos ωt
(σ = 0, s = ±j ω)
An exponentially varying sinusoid eσt cos ωt
(s = σ ± jω)
The Exponential Function est (3)
The Complex Frequency Plane s = σ + jω
s on y axis
+jω
s on right of y axis
-σ
+σ
s on left of y axis
-jω
s on x axis
Even and Odd functions (1)
• A real function xe(t) is said to be even function of t if
xe(t) = xe(-t)
• A real function xo(t) is said to be an odd function t if
xo(t) = - xo(-t)
Even and Odd functions (2)
Even and odd functions have the following properties:
• Even x Odd = Odd
• Odd x Odd = Even
• Even x Even = Even
Every signal x(t) can be expressed as a sum of even and
odd components because:
Even and Odd functions (3)
• Consider the causal exponential function