The Discrete-Time Fourier Transform (DTFT)

When the system is linear and discrete time-invariant (LIT system), only one representation stands out as the most useful. It is called The Discrete-Time Fourier Transform (DTFT) and is based on the complex exponential signal set {e^jωn}.

THE  DISCRETE-TIME FOURIER TRANSFORM (DTFT)

If x_[n] is absolutely summable, that is:

Then, the Discrete-Time Fourier Transform of x_[n] is given by:

Example 1.

Determine the DTFT of x_[n]:

Solution.

Since X[e^jω] is a complex-valued function, will have to plots its magnitude and its angle (or the real part and imaginary part) with respect to ω separately to visually describe X[e^jω]. Now ω is a real variable between –∞ and +∞, which would mean that we can plot only a part of the X[e^jω] function. Using two important properties of the DTFT, we can reduce this domain to the [0,π] interval for real-valued sequences.

The following Matlab script allows us to plot every part of X(e^jω) of example 1:

w=[0:1:500]*pi/500; X=exp(i*w)./(exp(i*w)-0.5*ones(1,501));
magX=abs(X);angX=angle(X);
plot(w/pi,magX); grid
xlabel(‘Frequency in pi units’); ylabel(‘Magnitude’);
title(‘Magnitude Part’)

This script yields:

w=[0:1:500]*pi/500; X=exp(i*w)./(exp(i*w)-0.5*ones(1,501));
angX=angle(X);
plot(w/pi,angX); grid
xlabel(‘Frequency in pi units’); ylabel(‘Radians’);
title(‘Angle Part’)

This script yields:

Example 2.

Determine the DTFT of the following finite-duration sequence:

Solution.

Two important properties

The following two properties are essential for DTFT analysis:

Some common DTFT Pairs

Derived from the previous properties, the DTFT of the following sequences, Table 1, are very useful:

Properties of the DTFT

We now present the complete properties of the DTFT in Table 2:

These properties will be of remarkable value for the next application of the DTFT: The z-Transformation.

We early stated that the Fourier Transform representation is the most useful signal representation for LTI systems. That is true due to the following reason:

RESPONSE TO A COMPLEX EXPONENTIAL ejωon

Let  be the input to an LTI system represented by the impulse response h_[n]:

Then:

Definition: The FREQUENCY RESPONSE

FREQUENCY RESPONSE: The Discrete-Time Fourier Transform of an impulse response is called The Frequency Response (or The Transfer Function) of an LTI system and is denoted by:

In consequence, if x_[n] is the input to an LTI system:

The system can be represented by:

and the output is as follows:

… Next: The Frequency Response of an LTI system

Source:

• Digital Signal Processing Using Matlab, 3erd ed
• Fundamentos_de_Señales_y_Sistemas_usando la Web y Matlab
• Oppenheim – Señales y Sistemas
• Análisis de Sistemas Lineales Asistido con Scilab – Un Enfoque desde la Ingeniería Eléctrica.

You can also be interested in:

• What is DSP ?- Digital Signal Processing
• Matlab code for DSP – Introduction
• Elementary sequences in Digital Signal Processing (Versión en Español)
• The Geometric Series in DSP (Versión en Español)
• The sinusoidal function in discrete-time (Versión en Español)
• Convolution in Discrete-Time in Matlab (Versión en Español)
• Solving discrete-time differential equations with Matlab (Versión en Español)
• Digital Filters – Moving Average Filters (Versión en Español)

Literature review by:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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