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Continuous Time Fourier Transform to Discrete Time by Sampling

Nasser M. Abbasi

April 7 2010 compiled on — Wednesday July 06, 2016 at 08:24 AM
The main purpose of this Demonstration is to illustrate the relation between the continuous-time Fourier transform (CTFT) of a continuous time signal xa(t)  and the discrete time Fourier transform (DTFT) of the discrete signal x(n)  generated from xa(t)  by sampling.

This Demonstration also illustrates the use of different algorithms to reconstruct the signal xa(t)  from the signal x(n)  .

The time domain signal xa(t)  is made of two harmonics. You can vary the amplitude and the frequency of each harmonic as well as the sampling frequency, the duration of x (t)
 a  , and the delay time. The spectrum of xa(t)  and of x(n)  are also displayed (the magnitude and the phase spectra). A number of plotting options are available to help analyze the results generated.

The following important relation between the CTFT and the DTFT is observed: The CTFT is an aperiodic and continuous function. The DTFT is also a continuous function, but it is a periodic function with a period of 2 π  . 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  of the magnitude spectrum of x (t)
 a  , where T  is the sampling period. The units of the DTFT are radians, while the units of the CTFT are in radians per second.