WOLFRAM|DEMONSTRATIONS PROJECT

From Continuous- to Discrete-Time Fourier Transform by Sampling Method

The purpose of this Demonstration is to show the relation between the continuous-time Fourier transform (CTFT) of a signal
x
a
(t)
and the corresponding discrete-time Fourier transform (DTFT) of the signal
x(n)
generated from
x
a
(t
) by sampling.
This Demonstration also illustrates the use of different algorithms to reconstruct
x
a
(t)
from
x(n)
.
The time domain signal
x
a
(t)
consists of two harmonics. You can vary the amplitude and the frequency of each harmonic as well as the sampling frequency, the duration of
x
a
(t)
, and the delay time. The magnitude and the phase spectra of
x
a
(t)
and of
x(n)
are also displayed. A number of plotting options are available to help analyze the results generated.
It is observed that if the CTFT is an aperiodic and continuous function, the DTFT is a continuous function, periodic in
2π
. The maximum frequency present in the DTFT is
π
in radians (or
1
2
in cycles). In addition, there is a scaling effect: the magnitude spectrum of
x(n)
is
1
T
smaller than the magnitude spectrum of
x
a
(t)
, where
T
is the sampling period. The units of the DTFT are radians, while the units of the CTFT are in radians per second.