University of Rhode Island Department of Electrical and Computer Engineering
ELE 436: Communication Systems
FFT Tutorial
1
Getting to Know the FFT
What is the FFT?
FFT = Fast Fourier Transform. The FFT is a faster version of the Discrete
Fourier Transform (DFT). The FFT utilizes some clever algorithms to do the same thing as the
DTF, but in much less time.
Ok, but what is the DFT?
The DFT is extremely important in the area of frequency (spectrum)
analysis because it takes a discrete signal in the time domain and transforms that signal into its
discrete frequency domain representation. Without a discretetime to discretefrequency transform
we would not be able to compute the Fourier transform with a microprocessor or DSP based system.
It is the speed and discrete nature of the FFT that allows us to analyze a signal’s spectrum with
Matlab or in realtime on the SR770
2
Review of Transforms
Was the DFT or FFT something that was taught in ELE 313 or 314?
No. If you took
ELE 313 and 314 you learned about the following transforms:
Laplace Transform:
x
(
t
)
⇔
X
(
s
)
where
X
(
s
) =
∞
R
∞
x
(
t
)
e

st
dt
ContinuousTime Fourier Transform:
x
(
t
)
⇔
X
(
jω
)
where
X
(
jω
) =
∞
R
∞
x
(
t
)
e

jωt
dt
z Transform:
x
[
n
]
⇔
X
(
z
)
where
X
(
z
) =
∞
∑
n
=
∞
x
[
n
]
z

n
DiscreteTime Fourier Transform:
x
[
n
]
⇔
X
(
e
j
Ω
)
where
X
(
e
j
Ω
) =
∞
∑
n
=
∞
x
[
n
]
e

j
Ω
n
The
Laplace transform
is used to to find a pole/zero representation of a continuoustime signal or
system,
x
(
t
), in the splane. Similarly, The
z transform
is used to find a pole/zero representation
of a discretetime signal or system,
x
[
n
], in the zplane.
The continuoustime Fourier transform
(CTFT)
can be found by evaluating the Laplace trans
form at
s
=
jω
. The discretetime Fourier transform
(DTFT)
can be found by evaluating the z
transform at
z
=
e
j
Ω
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Harris,J
 Digital Signal Processing, DFT, FFT

Click to edit the document details