7.1 Definition and Properties of DTFT

Discrete-Time Fourier Transform: Definition and Properties

The Discrete-Time Fourier Transform (DTFT) converts a discrete-time signal into a continuous function of frequency. It's the primary tool for understanding what frequencies are present in a sampled signal and how discrete-time systems (like digital filters) behave across frequency.

This section covers the formal definition, the key properties you'll use repeatedly, the DTFT of common signals, and how the transform applies to signal and system analysis.

Discrete-Time Fourier Transform

Definition and Inverse

The DTFT takes a discrete-time signal $x[n]$ and produces a continuous, complex-valued function of angular frequency $\omega$:

$X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \, e^{-j\omega n}$

Each value of $X(\omega)$ tells you how much of the frequency $\omega$ is present in the signal. The complex exponential $e^{-j\omega n}$ acts as the "probe" at each frequency.

To recover the original signal from its frequency-domain representation, you use the inverse DTFT:

$x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega) \, e^{j\omega n} \, d\omega$

Notice the integration is only over one period, from $-\pi$ to $\pi$. That's because $X(\omega)$ is periodic with period $2\pi$:

$X(\omega) = X(\omega + 2\pi)$

This periodicity comes directly from the fact that $e^{-j\omega n} = e^{-j(\omega + 2\pi)n}$ for integer $n$. It's a fundamental difference from the continuous-time Fourier transform, where the spectrum is not periodic.

Existence condition: The DTFT exists (converges) for any signal that is absolutely summable:

$\sum_{n=-\infty}^{\infty} |x[n]| < \infty$

Signals that don't satisfy this condition (like the unit step or a pure sinusoid) can still have DTFTs, but they require impulse functions $\delta(\omega)$ in the frequency domain.

Properties

These properties are the workhorses of DTFT analysis. Rather than computing the full summation every time, you can use these to build up transforms of complex signals from simpler ones.

• Linearity: If $y[n] = a\,x[n] + b\,z[n]$, then $Y(\omega) = a\,X(\omega) + b\,Z(\omega)$. Scaling and adding in time maps directly to scaling and adding in frequency.
• Time shift: If $y[n] = x[n - n_0]$, then $Y(\omega) = e^{-j\omega n_0} X(\omega)$. Delaying a signal by $n_0$ samples multiplies its DTFT by a complex exponential. The magnitude spectrum stays the same; only the phase changes. This is worth remembering: shifting in time doesn't alter what frequencies are present, just their relative timing.
• Frequency shift: If $y[n] = x[n]\,e^{j\omega_0 n}$, then $Y(\omega) = X(\omega - \omega_0)$. Multiplying by a complex exponential in time slides the entire spectrum by $\omega_0$. This is the basis of modulation.
• Conjugation: If $y[n] = x^*[n]$, then $Y(\omega) = X^*(-\omega)$. For real-valued signals (where $x[n] = x^*[n]$), this gives you the important conjugate symmetry property: $X(\omega) = X^*(-\omega)$. That means the magnitude spectrum is even and the phase spectrum is odd.
• Time reversal: If $y[n] = x[-n]$, then $Y(\omega) = X(-\omega)$. Flipping the signal in time flips the spectrum in frequency. For a real-valued signal, since $X(-\omega) = X^*(\omega)$, time reversal is equivalent to conjugating the spectrum.
• Convolution: If $y[n] = x[n] * h[n]$, then $Y(\omega) = X(\omega)\,H(\omega)$. Convolution in time becomes multiplication in frequency. This single property is why the DTFT is so useful for system analysis: finding a system's output reduces to multiplying two frequency-domain functions.

DTFT Applications

Signal Analysis

The DTFT reveals the spectral content of a discrete-time signal. By examining $|X(\omega)|$, you can identify dominant frequency components, measure bandwidth, and detect periodicity.

The DTFT is also central to understanding sampling and aliasing. When a continuous-time signal is sampled, its DTFT is a scaled, periodic version of the original continuous-time spectrum. If the sampling rate was too low, the periodic copies overlap, and you see aliasing directly in the DTFT.

For system output analysis, the convolution property makes things straightforward: if you know the input spectrum $X(\omega)$ and the system's frequency response $H(\omega)$, the output spectrum is just $Y(\omega) = X(\omega)\,H(\omega)$.

System Analysis

The frequency response of a discrete-time LTI system is the DTFT of its impulse response $h[n]$:

$H(\omega) = \sum_{n=-\infty}^{\infty} h[n] \, e^{-j\omega n}$

This function $H(\omega)$ tells you exactly how the system modifies each frequency. Its magnitude $|H(\omega)|$ gives the gain at each frequency, and its phase $\angle H(\omega)$ gives the phase shift.

Filter design is built on this idea. A low-pass filter has $|H(\omega)| \approx 1$ for small $|\omega|$ and $|H(\omega)| \approx 0$ near $\pi$. A high-pass filter does the opposite. By shaping $H(\omega)$ to the desired profile, you design the filter's impulse response accordingly.

DTFT of Common Signals

Basic Signals

These are reference transforms you should know. They come up constantly when building more complex transforms using DTFT properties.

• Unit impulse $\delta[n]$: Since $\delta[n]$ is nonzero only at $n = 0$, the summation collapses to a single term:

$X(\omega) = 1 \quad \text{for all } \omega$

A single impulse contains all frequencies equally. This is a flat spectrum.

• Causal exponential $x[n] = a^n u[n]$, where $|a| < 1$: The absolute summability condition is satisfied because the geometric series converges, giving:

$X(\omega) = \frac{1}{1 - a\,e^{-j\omega}}$

When $a$ is real and positive, this spectrum peaks at $\omega = 0$ (low-pass character). When $a$ is real and negative, it peaks at $\omega = \pi$ (high-pass character).

• Unit step $u[n]$: This signal is not absolutely summable, so its DTFT requires an impulse term:

$U(\omega) = \pi\,\delta(\omega) + \frac{1}{1 - e^{-j\omega}}$

The $\pi\,\delta(\omega)$ term accounts for the DC (zero-frequency) content of the step. This expression is understood as periodic with period $2\pi$, so $\delta(\omega)$ here is a periodic impulse train.

Periodic and Finite-Duration Signals

• Cosine $x[n] = \cos(\omega_0 n)$: A pure cosine produces two impulses in the frequency domain:

$X(\omega) = \pi\left[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)\right]$

This follows directly from writing $\cos(\omega_0 n) = \frac{1}{2}(e^{j\omega_0 n} + e^{-j\omega_0 n})$ and applying the frequency-shift property to the known DTFT of a constant signal.

• Rectangular pulse of length $N$ ($x[n] = 1$ for $0 \le n \le N-1$, zero otherwise): Summing the geometric series gives:

$X(\omega) = e^{-j\omega(N-1)/2} \cdot \frac{\sin(\omega N / 2)}{\sin(\omega / 2)}$

The ratio $\frac{\sin(\omega N / 2)}{\sin(\omega / 2)}$ is called the Dirichlet kernel. Its main lobe width is $\frac{4\pi}{N}$, so longer pulses produce narrower main lobes (better frequency concentration). The exponential prefactor accounts for the linear phase from the pulse not being centered at $n = 0$.

• General finite-duration signals (windowed sinusoids, truncated exponentials, etc.) can be computed by applying the DTFT definition directly or by combining the transforms above using linearity, time shift, and frequency shift properties.