Graphical Convolution Summary
Given h[n] and x[n], we want to find y[n] for all n, where
D-T Convolution:
y [ n ]=
h [ i ] x [ n i ]
i =
To find y[n] graphically:
1. Fix n.
2. Form h[i]
3. Form x[-i]
4. Form x[n i]
5. Compute the product h[i] x[n i] for a
Fourier Transform
jt
X = x t e
dt
A sufficient condition for the existence of
the Fourier Transform of x(t) includes:
x t dt
Generalized FT
This section allows us to apply FT to an even broader class of signals that
includes some periodic signals and o
Fourier Series Expansion
Fourier Series Expansion
T = 2/0
x t =
k =
ck e
jk t
0
0 = fundamental frequency (rad/sec)
Fourier Series
(Complex Exponential
Form)
x t = a0 [ a k cos k0 t b k sin k0 t ]
k =1
Fourier Series
(Trigonometric Form)
x t = A0 A k cos
Step and Ramp Functions
These are common textbook signals but are also common test signals,
especially in control systems.
Unit Step Function u(t)
cfw_
u t = 1, t 0
0, t 0
u(t)
.
1
.
t
Note: A step of height A can be made from Au(t)
In system analysis, wh
Fourier Transform Properties
We will consider various properties of the Fourier Transform
that will help in finding the FT of various signals.
X = x t e jt dt
1
jt
x t = X e d
2
1. Linearity (Supremely Important)
If
x t X & y t Y
then [ ax t by t ] [ aX
y [ n ] = x [ n ] 2 x [ n 2 ]
Causal or non-causal?
It is causal since the output is expressed only in
terms of current or past versions of the input.
Memory or memoryless?
It has memory since the current output includes a
past version of the input.
Lin
Fourier Series
x t =
k =
ck e
jk t
0
jk t
1 t T
0
c k = t x t e
dt
T
0
0
1
2
Circular Convolution
3
Note the limits on
the circular
convolution
integral!
What are the
Fourier Series
coefficients of
the new
periodic signal?
Products of Periodic Functions
The Sampling Theorem
The Connection Between
Continuous and Discrete Time!
This is also called the NyquistShannon Sampling
Theorem or the Nyquist Sampling Theorem (implied by
Harry Nyquist in 1928 and proven by Claude Shannon in
1949.)
These overheads were
Fourier Analysis
(Chapter 3)
These overheads were originally developed by Mark Fowler at Binghamton University,
State University of New York.
Ch. 3: Fourier Series & Fourier Transform
(This chapter is for C-T case only)
i.e., sinusoids
3.1 Representation
The Fourier Transform
4.3 Fourier Transform
Recall: Fourier Series represents a periodic signal as a sum of sinusoids
or complex sinusoids e
jk 0 t
Note: Because the FS uses harmonically related frequencies k0 , it can only create
periodic signals
Q: Can
Fourier Analysis
We considered sums of sinusoids with
frequencies that are multiples of some
fundamental frequency, 0
N
k =0 Ak cos ( k 0 t +k )
k = 1, 2, 3, . . .
0 > 0
We found that in this case the result is
always periodic regardless of the values
of
Convolution Properties
These are things you can exploit to make it easier to solve problems
x [ n ]h [ n ]= h [ n ] x [ n ]
1.Commutativity
You can choose which signal to flip
2. Associativity
x [ n ] v [ n ] w [ n ] = x [ n ]v [ n ] w [ n ]
Can change