19.2 Fourier transform for aperiodic signals

Fourier Transform for Aperiodic Signals

Definition and Properties of Aperiodic Signals

Fourier series work great for periodic signals, but most real-world signals (a single clap, a transient voltage spike, a one-time pulse) don't repeat. These are aperiodic signals, and analyzing their frequency content requires the Fourier transform instead.

• An aperiodic signal $x(t)$ is any signal that does not repeat periodically over time.
• Common examples: a single rectangular pulse, a decaying exponential $e^{-at}u(t)$, or a chirp signal whose frequency sweeps over time.
• Unlike periodic signals (which have discrete frequency components at harmonics), aperiodic signals have a continuous frequency spectrum. Every frequency contributes, not just integer multiples of a fundamental.
• Aperiodic signals carry finite total energy, calculated as:

$E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$

This finite-energy requirement is what distinguishes signals suited for the Fourier transform from periodic signals, which have finite power but infinite energy.

The Fourier Transform

The Fourier transform decomposes an aperiodic time-domain signal into a continuous spectrum of frequencies. For a signal $x(t)$, the transform is:

$X(f) = \int_{-\infty}^{\infty} x(t) \, e^{-j2\pi ft} \, dt$

• $X(f)$ is the frequency-domain representation of the signal.
• $f$ is frequency (in Hz).
• $j$ is the imaginary unit ($j^2 = -1$).

You can think of this integral as correlating $x(t)$ with a complex exponential at each frequency $f$. Where the signal has strong content at a particular frequency, $X(f)$ will be large there.

Existence condition: The transform exists when $x(t)$ is absolutely integrable:

$\int_{-\infty}^{\infty} |x(t)| \, dt < \infty$

Frequency Spectrum and Continuous Spectrum

Because $X(f)$ is generally complex-valued, it's common to split it into two parts:

• Magnitude spectrum $|X(f)|$: Shows the amplitude of each frequency component. This tells you which frequencies are present and how strong they are.
• Phase spectrum $\angle X(f)$: Shows the phase shift of each frequency component. This tells you the relative timing between different frequency components.

Both pieces are needed to fully reconstruct the original signal. Two signals can share the same magnitude spectrum but sound or look completely different because of their phase spectra.

Inverse Fourier Transform

Definition and Properties of the Inverse Fourier Transform

The inverse Fourier transform takes you back from the frequency domain to the time domain, reconstructing $x(t)$ from $X(f)$:

$x(t) = \int_{-\infty}^{\infty} X(f) \, e^{j2\pi ft} \, df$

Notice the symmetry with the forward transform. The only differences are the sign of the exponent ($+j$ instead of $-j$) and that you're integrating over frequency instead of time.

The inverse transform exists when $X(f)$ is absolutely integrable: $\int_{-\infty}^{\infty} |X(f)| \, df < \infty$.

Together, the forward and inverse transforms form a transform pair: you can move freely between domains without losing information.

Dirac Delta Function and Its Properties

The Dirac delta function $\delta(t)$ is a mathematical idealization that shows up constantly in Fourier analysis. It represents an infinitely narrow, infinitely tall pulse with unit area:

$\int_{-\infty}^{\infty} \delta(t) \, dt = 1$

Its most useful property is the sifting property: for any continuous function $g(t)$,

$\int_{-\infty}^{\infty} g(t) \, \delta(t - t_0) \, dt = g(t_0)$

This "picks out" the value of $g$ at $t_0$.

Two key Fourier transform pairs involving the delta function:

• $\mathcal{F}\{\delta(t)\} = 1$ (an impulse in time has equal content at all frequencies)
• $\mathcal{F}^{-1}\{1\} = \delta(t)$ (a constant spectrum in frequency corresponds to a single impulse in time)

These pairs are useful for deriving other transforms and for modeling idealized impulse inputs to systems.

Convolution Theorem and Its Application

Convolution describes how a linear time-invariant (LTI) system processes an input signal. The convolution of $x(t)$ and $h(t)$ is:

$(x * h)(t) = \int_{-\infty}^{\infty} x(\tau) \, h(t - \tau) \, d\tau$

Computing this integral directly can be tedious. The convolution theorem provides a shortcut:

$\mathcal{F}\{x(t) * h(t)\} = X(f) \cdot H(f)$

In other words, convolution in the time domain becomes simple multiplication in the frequency domain. This is one of the most practically important results in signal processing.

To find the output of an LTI system using this theorem:

1. Compute $X(f) = \mathcal{F}\{x(t)\}$ (transform the input).
2. Compute $H(f) = \mathcal{F}\{h(t)\}$ (transform the impulse response).
3. Multiply: $Y(f) = X(f) \cdot H(f)$.
4. Inverse transform: $y(t) = \mathcal{F}^{-1}\{Y(f)\}$.

The reverse also holds: multiplication in time corresponds to convolution in frequency.

Time-Frequency Duality

Properties and Implications of Time-Frequency Duality

Time-frequency duality refers to the deep symmetry between the time and frequency domains. Operations you perform in one domain have direct counterparts in the other. Recognizing these pairs saves a lot of work because you can often derive a new result by applying duality to one you already know.

Here are the most important duality relationships:

A few things to notice:

• Scaling tradeoff: Compressing a signal in time (making it shorter) spreads its spectrum wider in frequency, and vice versa. A very narrow pulse has a very wide bandwidth. This is a fundamental constraint you'll encounter repeatedly in communications and signal processing.
• Time shift → phase shift: Shifting a signal in time doesn't change which frequencies are present (the magnitude spectrum stays the same), it only changes when they arrive (the phase spectrum shifts linearly with frequency).
• Convolution ↔ multiplication: This pair is the convolution theorem from the previous section, stated in both directions.

These duality properties make the Fourier transform especially powerful for circuit and system analysis, because difficult operations in one domain often become straightforward operations in the other.