6. Discrete-Time Fourier Transform (DTFT)

What is DTFT?

The Discrete-Time Fourier Transform (DTFT) is a mathematical tool used to analyze discrete-time signals in the frequency domain. Unlike the usual Fourier Transform that works on continuous-time signals, the DTFT is specifically for signals defined at discrete time steps (like sequences).

It helps to understand the frequency content of signals that are discrete in time but can be infinite in duration.

6.1. Signals: Discrete-time and Aperiodic

• Discrete-time signal: A sequence x[n], where n is an integer representing time steps (e.g., n = ..., -2, -1, 0, 1, 2, ...).
• Aperiodic signal: A signal that does not repeat itself over time (no fixed period).

The DTFT is used for discrete-time, aperiodic signals, meaning the signal values are sampled at discrete time instants, and the signal doesn't repeat periodically.

6.2. Definition of DTFT

The DTFT of a discrete-time signal x[n] is defined as:

X(ω) = Σn=-∞∞ x[n] · e-jωn
  

• X(ω) is the frequency domain representation of the signal.
• ω is the angular frequency in radians per sample (continuous variable ranging from -π to π).
• j = √-1, the imaginary unit.
• The sum runs over all integer values of n.

6.3. Inverse DTFT

To recover the original discrete-time signal x[n] from its DTFT X(ω):

x[n] = (1/2π) ∫-ππ X(ω) · ejωn dω
  

• The integral runs over one period of the DTFT in frequency, from -π to π.
• This shows that the DTFT is invertible, meaning you can go back and forth between time and frequency domains.

6.4. Key Properties of DTFT

6.5. Relationship to Z-Transform

The Z-transform of a discrete-time signal x[n] is:

X(z) = Σn=-∞∞ x[n] · z-n
  

Here, z is a complex number: z = r · ejω, where r is radius and ω is angle (frequency).

The DTFT is a special case of the Z-transform, evaluated on the unit circle |z| = 1:

X(ω) = X(z) |z = ejω = Σn=-∞∞ x[n] · e-jωn
  

Interpretation:

• The Z-transform gives a more general frequency representation for signals, including stability and causality info (when r ≠ 1).
• DTFT focuses only on frequency content by setting r = 1.
• This means DTFT shows how different frequencies are present in a signal, assuming the signal has no growth or decay (stable on the unit circle).

6.6. Example: DTFT of a Simple Signal

Example: Find the DTFT of

x[n] = {
  1, n = 0,1,2
  0, otherwise
}
  

Step 1: Write the definition

X(ω) = Σn=-∞∞ x[n] · e-jωn = Σn=02 e-jωn
  

(because x[n] = 0 outside 0 to 2)

Step 2: Calculate the sum

X(ω) = 1 + e-jω + e-j2ω
  

This can also be expressed using the formula for a finite geometric series:

X(ω) = (1 - e-j3ω) / (1 - e-jω)
  

6.7. Interpretation of the Example

• X(ω) is periodic in ω with period 2π.
• The magnitude |X(ω)| tells you how much of each frequency component exists in x[n].
• Because x[n] is just three samples of 1, its frequency content spreads over all frequencies with some periodic pattern.

6.8. Visualization

Below is a graph plotting the magnitude |X(ω)| over frequency ω from -π to π.

6.9. Why is DTFT Important?

• Analyzes frequency components of discrete signals.
• Helps design digital filters.
• Bridges between time and frequency analysis in digital signal processing.
• Fundamental for understanding spectral behavior of sampled data.

Summary