6.2 Discrete-time Fourier transform (DTFT)

The Discrete-Time Fourier Transform (DTFT) is a powerful tool for analyzing discrete-time signals in the frequency domain. It converts time-domain signals into continuous functions of frequency, revealing their spectral characteristics and frequency content.

Understanding the DTFT is crucial for signal processing and system analysis. It helps engineers interpret magnitude and phase spectra, apply properties like periodicity and symmetry, and grasp the relationship between time-domain shifts and frequency-domain phase changes.

Discrete-Time Fourier Transform (DTFT)

Definition of DTFT and inverse

• Represents discrete-time signals in the frequency domain by converting a discrete-time signal $x[n]$ into a continuous function of frequency $X(e^{j\omega})$
• DTFT formula: $X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n}$
• Inverse DTFT converts a frequency-domain representation back to a discrete-time signal using the formula $x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega})e^{j\omega n} d\omega$

Application of DTFT for analysis

• Provides insight into the frequency content and spectral characteristics of a discrete-time signal
• Steps to apply DTFT:
  1. Calculate $X(e^{j\omega})$ using the DTFT formula for a given discrete-time signal $x[n]$
  2. Simplify the resulting expression using properties of complex exponentials and summations
  3. Analyze the frequency-domain representation to understand the signal's spectral content (e.g., dominant frequencies, bandwidth)

Interpretation of magnitude and phase

• DTFT expressed in polar form: $X(e^{j\omega}) = |X(e^{j\omega})|e^{j\angle X(e^{j\omega})}$
  • Magnitude spectrum $|X(e^{j\omega})|$ shows the amplitude of each frequency component
  • Phase spectrum $\angle X(e^{j\omega})$ shows the phase shift of each frequency component
• Magnitude spectrum indicates the relative strength of different frequencies in the signal (e.g., low-frequency components, high-frequency components)
• Phase spectrum provides information about the time delay or shift of each frequency component (e.g., linear phase, nonlinear phase)

Relationship between DTFT and z-transform

• DTFT is a special case of the z-transform evaluated on the unit circle ($z = e^{j\omega}$) in the z-plane
• z-transform formula: $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$
• z-transform allows for the analysis of signals with regions of convergence (ROC) beyond the unit circle (e.g., causal, anticausal systems)
• DTFT is useful for analyzing the frequency content of stable, causal systems with ROC including the unit circle

Properties of DTFT

• Periodicity: DTFT is periodic with a period of $2\pi$, i.e., $X(e^{j\omega}) = X(e^{j(\omega + 2\pi)})$ for all $\omega$
• Symmetry properties for real-valued $x[n]$:
  • Magnitude spectrum is even symmetric: $|X(e^{j\omega})| = |X(e^{-j\omega})|$
  • Phase spectrum is odd symmetric: $\angle X(e^{j\omega}) = -\angle X(e^{-j\omega})$
• Linearity: DTFT is a linear operation, i.e., if $x_1[n] \leftrightarrow X_1(e^{j\omega})$ and $x_2[n] \leftrightarrow X_2(e^{j\omega})$, then $ax_1[n] + bx_2[n] \leftrightarrow aX_1(e^{j\omega}) + bX_2(e^{j\omega})$
• Time shifting: A shift in the time domain corresponds to a phase shift in the frequency domain, i.e., if $x[n] \leftrightarrow X(e^{j\omega})$, then $x[n-n_0] \leftrightarrow e^{-j\omega n_0}X(e^{j\omega})$