The Discrete-Time Fourier Transform (DTFT) is a powerful tool for analyzing discrete-time signals in the frequency domain. It transforms time-domain signals into continuous frequency representations, revealing crucial spectral characteristics and enabling various signal processing applications.

Understanding the DTFT's definition, properties, and applications is essential for signal analysis and system design. This knowledge allows us to manipulate signals, study frequency responses, and develop effective filters for diverse engineering and scientific applications.

Discrete-Time Fourier Transform

Definition and Inverse

• The Discrete-Time Fourier Transform (DTFT) represents discrete-time signals in the frequency domain using the formula $X(ω) = Σ[n=-∞ to ∞] x[n]e^(-jωn)$, where $x[n]$ denotes the discrete-time signal and $ω$ represents the angular frequency
• The inverse DTFT recovers the original discrete-time signal from its frequency-domain representation using the formula $x[n] = (1/2π) ∫[ω=-π to π] X(ω)e^(jωn) dω$
• The DTFT yields a continuous function of frequency, despite the original signal being discrete in time
• The DTFT exists for any discrete-time signal that is absolutely summable, satisfying the condition $Σ[n=-∞ to ∞] |x[n]| < ∞$
• The DTFT exhibits periodicity with a period of $2π$, expressed as $X(ω) = X(ω + 2π)$

Properties

• Linearity property states that the DTFT of a linear combination of signals equals the linear combination of their individual DTFTs, i.e., if $y[n] = ax[n] + bz[n]$, then $Y(ω) = aX(ω) + bZ(ω)$
• Time-shifting property indicates that delaying a signal in time by $n_0$ samples results in a phase shift in the frequency domain, i.e., if $y[n] = x[n-n_0]$, then $Y(ω) = e^(-jωn_0) X(ω)$
• Frequency-shifting property shows that multiplying a signal by a complex exponential in time leads to a frequency shift in the DTFT, i.e., if $y[n] = x[n]e^(jω_0n)$, then $Y(ω) = X(ω-ω_0)$
• Conjugation property demonstrates that the DTFT of the complex conjugate of a signal equals the complex conjugate of the DTFT of the original signal, evaluated at the negative frequency, i.e., if $y[n] = x*[n]$, then $Y(ω) = X*(-ω)$
• Time-reversal property reveals that reversing a signal in time results in the complex conjugate of its DTFT, i.e., if $y[n] = x[-n]$, then $Y(ω) = X(-ω)$
• Convolution property establishes that the convolution of two signals in the time domain corresponds to multiplication of their DTFTs in the frequency domain, i.e., if $y[n] = x[n] h[n]$, then $Y(ω) = X(ω)H(ω)$

DTFT Applications

Signal Analysis

• The DTFT helps determine the frequency content of a discrete-time signal, enabling the identification of dominant frequencies, bandwidth, and other spectral characteristics
• The DTFT can be used to study the effects of sampling and aliasing in discrete-time signals and systems
• The DTFT allows for the analysis of the output of a discrete-time system by multiplying the DTFTs of the input signal and the system's impulse response, leveraging the convolution property

System Analysis

• The DTFT can be employed to analyze the frequency response of discrete-time systems, such as filters, by examining the DTFT of the system's impulse response
• The convolution property of the DTFT facilitates the analysis of the output of a discrete-time system by multiplying the DTFTs of the input signal and the system's impulse response
• The DTFT enables the design and analysis of various discrete-time systems, including low-pass, high-pass, and band-pass filters, by manipulating the system's frequency response

DTFT of Common Signals

Basic Signals

• The DTFT of the unit impulse signal, $δ[n]$, equals 1 for all frequencies, i.e., $Δ(ω) = 1$
• The DTFT of the unit step signal, $u[n]$, is given by $U(ω) = πδ(ω) + 1/(1-e^(-jω))$
• The DTFT of the exponential signal, $x[n] = a^n u[n]$, where $|a| < 1$, is given by $X(ω) = 1/(1 - ae^(-jω))$

Periodic and Finite-Duration Signals

• The DTFT of the sinusoidal signal, $x[n] = cos(ω_0n)$, is given by $X(ω) = π[δ(ω-ω_0) + δ(ω+ω_0)]$
• The DTFT of the rectangular pulse signal, $x[n] = rect[n/N]$, is given by $X(ω) = sin(ωN/2) / sin(ω/2)$
• The DTFT of a finite-duration signal, such as a windowed sinusoid or a truncated exponential, can be obtained by applying the DTFT definition and properties to the specific signal