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 discrete-time to discrete-frequency 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 real-time 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
Continuous-Time Fourier Transform: x(t) ⇔ X(jω) where X(jω) =
∞
R
−∞
x(t)e
−jωt
dt
z Transform: x[n] ⇔ X(z) where X(z) =
∞
P
n=−∞
x[n]z
−n
Discrete-Time Fourier Transform: x[n] ⇔ X(e
jΩ
) where X(e
jΩ
) =
∞
P
n=−∞
x[n]e
−jΩn
The Laplace transform is used to to find a pole/zero representation of a continuous-time signal or
system, x(t), in the s-plane. Similarly, The z transform is used to find a pole/zero representation
of a discrete-time signal or system, x[n], in the z-plane.
The continuous-time Fourier transform (CTFT) can be found by evaluating the Laplace trans-
form at s = jω. The discrete-time Fourier transform (DTFT) can be found by evaluating the z
transform at z = e
jΩ
.
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