1.2 Continuous-time Fourier transform

The continuous-time Fourier transform (CTFT) decomposes continuous-time signals into their frequency components, revealing the spectral content of any signal you're working with. It's the core analytical tool connecting time-domain and frequency-domain representations, and nearly every technique in advanced signal processing builds on it.

This guide covers the CTFT definition and inverse, its key properties, transforms of basic and periodic signals, convolution/multiplication duality, the relationship to the Laplace transform, and sampling theory.

Definition of continuous-time Fourier transform

The CTFT takes a continuous-time signal and produces a complex-valued function of frequency that tells you exactly which frequency components are present and with what amplitude and phase.

Mathematical representation

The CTFT of a signal $x(t)$ is defined as:

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

• $f$ is the frequency variable in Hertz (Hz)
• $j$ is the imaginary unit ($j^2 = -1$)
• $X(f)$ is complex-valued: its magnitude $|X(f)|$ gives the amplitude spectrum, and its angle $\angle X(f)$ gives the phase spectrum

The integral essentially correlates $x(t)$ with a complex exponential at each frequency $f$. Wherever the signal contains energy at that frequency, the integral produces a large value.

Fourier transform pair

The CTFT and the original signal form a Fourier transform pair, written as $x(t) \leftrightarrow X(f)$. This notation means the mapping is one-to-one: a signal has a unique spectrum, and a spectrum corresponds to a unique signal. You can work in whichever domain is more convenient and convert back without losing information.

Inverse Fourier transform

To recover the time-domain signal from its spectrum, use the inverse CTFT:

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

Notice the only differences from the forward transform: the sign in the exponent flips from $-j$ to $+j$, and there's no extra scaling factor (in the Hz-frequency convention). This symmetry between the forward and inverse transforms is a recurring theme.

Properties of Fourier transform

These properties let you predict how operations in one domain affect the other domain, often turning difficult time-domain problems into straightforward frequency-domain algebra.

Linearity

The CTFT obeys superposition. If $x_1(t) \leftrightarrow X_1(f)$ and $x_2(t) \leftrightarrow X_2(f)$, then:

$a_1 x_1(t) + a_2 x_2(t) \leftrightarrow a_1 X_1(f) + a_2 X_2(f)$

This means you can analyze complex signals by breaking them into simpler pieces, transforming each piece separately, and adding the results.

Time shifting

$x(t - t_0) \leftrightarrow X(f) e^{-j2\pi f t_0}$

Delaying a signal by $t_0$ seconds doesn't change the magnitude spectrum at all. It only adds a linear phase term $-2\pi f t_0$. This is why a time delay shows up purely as a phase shift in the frequency domain.

Frequency shifting

$x(t) e^{j2\pi f_0 t} \leftrightarrow X(f - f_0)$

Multiplying by a complex exponential in time slides the entire spectrum by $f_0$ Hz. This is the mathematical basis of modulation: to move a baseband signal up to a carrier frequency, you multiply by a complex exponential (or equivalently, a cosine, which shifts the spectrum to both $+f_0$ and $-f_0$).

Time scaling

$x(at) \leftrightarrow \frac{1}{|a|} X\!\left(\frac{f}{a}\right)$

Compressing a signal in time ($|a| > 1$) spreads its spectrum wider and reduces its amplitude. Stretching a signal in time ($|a| < 1$) narrows the spectrum. This inverse relationship between time duration and spectral bandwidth is fundamental: you cannot have a signal that is both short in time and narrow in bandwidth.

Duality

If $x(t) \leftrightarrow X(f)$, then $X(t) \leftrightarrow x(-f)$.

Duality means that if you know a transform pair, you automatically get a second pair for free by swapping the roles of time and frequency. For example, since a rectangular pulse transforms to a sinc, duality tells you that a sinc-shaped time signal transforms to a rectangular spectrum.

Parseval's theorem

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

The total energy of a signal is the same whether you compute it in the time domain or the frequency domain. The quantity $|X(f)|^2$ is the energy spectral density, which tells you how the signal's energy is distributed across frequency. This is essential for spectral analysis and filter design, where you need to know how much energy sits in a particular frequency band.

Fourier transform of basic signals

These standard transform pairs serve as building blocks. Knowing them lets you quickly determine the spectrum of more complex signals using linearity and other properties.

Impulse function

$\delta(t) \leftrightarrow 1$

The Dirac delta function has a perfectly flat spectrum: equal contribution at every frequency. This makes physical sense because an infinitely short, infinitely tall pulse must contain all frequencies to produce such extreme localization in time. The delta function also characterizes LTI systems, since the output to $\delta(t)$ is the impulse response $h(t)$.

Step function

$u(t) \leftrightarrow \frac{1}{j2\pi f} + \frac{1}{2}\delta(f)$

The unit step function jumps from 0 to 1 at $t = 0$. Its spectrum has two parts: a $1/(j2\pi f)$ term that decays with frequency (reflecting the sharp transition), and an impulse at $f = 0$ representing the DC (constant) component. The DC impulse appears because the step function has a nonzero average value.

Rectangular pulse

$\text{rect}\!\left(\frac{t}{T}\right) \leftrightarrow T\,\text{sinc}(fT)$

where $\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$.

The rect function equals 1 for $|t| < T/2$ and 0 elsewhere. Its spectrum is a sinc function whose main lobe has a width of $2/T$ Hz. A shorter pulse (smaller $T$) produces a wider main lobe, illustrating the time-bandwidth tradeoff. Rectangular pulses appear constantly in communications (pulse shaping) and spectral analysis (windowing).

Triangular pulse

$\Lambda\!\left(\frac{t}{T}\right) \leftrightarrow T\,\text{sinc}^2\!\left(\frac{fT}{2}\right)$

The triangular pulse ramps linearly up to a peak and back down. Its spectrum is $\text{sinc}^2$, which decays faster than the plain sinc of the rectangular pulse. The smoother shape in time produces less spectral leakage, which is why triangular (Bartlett) windows are sometimes preferred over rectangular windows in spectral estimation.

Sinusoidal signal

$\cos(2\pi f_0 t) \leftrightarrow \frac{1}{2}\left[\delta(f - f_0) + \delta(f + f_0)\right]$

A pure cosine at frequency $f_0$ produces two impulses in the spectrum: one at $+f_0$ and one at $-f_0$. The negative-frequency impulse isn't a separate physical frequency; it's a mathematical consequence of representing a real-valued cosine using complex exponentials. This pair is the starting point for understanding how the Fourier transform represents any sum of sinusoids.

Fourier transform of periodic signals

Periodic signals repeat with period $T$, and their spectra are discrete: energy exists only at integer multiples of the fundamental frequency $f_0 = 1/T$.

Fourier series representation

A periodic signal $x(t)$ with period $T$ can be written as:

$x(t) = \sum_{n=-\infty}^{\infty} c_n \, e^{j2\pi n f_0 t}$

The complex Fourier coefficients are computed by:

$c_n = \frac{1}{T} \int_{0}^{T} x(t) \, e^{-j2\pi n f_0 t} \, dt$

Each coefficient $c_n$ encodes the amplitude and phase of the $n$-th harmonic. The magnitude $|c_n|$ tells you how strong that harmonic is, and $\angle c_n$ tells you its phase.

Relationship between Fourier series and Fourier transform

Taking the CTFT of a periodic signal yields:

$X(f) = \sum_{n=-\infty}^{\infty} c_n \, \delta(f - nf_0)$

The spectrum is a train of impulses at multiples of $f_0$, with each impulse weighted by the corresponding Fourier coefficient $c_n$. This connects the two frameworks: the Fourier series gives you the coefficients, and the Fourier transform places those coefficients as impulse weights on the frequency axis. For periodic signals, all energy is concentrated at discrete frequencies rather than spread across a continuum.

Convolution and multiplication properties

The duality between convolution and multiplication is one of the most practically useful results in signal processing. It turns the difficult integral operation of convolution into simple pointwise multiplication.

Convolution in time domain

The convolution of $x(t)$ and $h(t)$ is:

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

Convolution describes the output of an LTI system: if $x(t)$ is the input and $h(t)$ is the impulse response, the output is $y(t) = x(t) * h(t)$. Conceptually, you flip $h$, slide it across $x$, and integrate the product at each position.

Multiplication in frequency domain

The convolution theorem states:

$x(t) * h(t) \leftrightarrow X(f) \cdot H(f)$

Convolution in time becomes multiplication in frequency. The dual also holds:

$x(t) \cdot h(t) \leftrightarrow X(f) * H(f)$

Multiplication in time becomes convolution in frequency. This second form explains why modulating a signal (multiplying by a carrier) spreads its spectrum.

Applications in signal processing

• Filtering: To apply a filter with frequency response $H(f)$, multiply the input spectrum $X(f)$ by $H(f)$. This is equivalent to convolving $x(t)$ with the filter's impulse response $h(t)$, but multiplication is often easier to analyze and implement (especially via FFT).
• Modulation: Multiplying $x(t)$ by a carrier $\cos(2\pi f_c t)$ convolves $X(f)$ with the carrier's spectrum (two impulses), shifting the signal to $\pm f_c$. This is the basis of AM transmission.
• Deconvolution: If you know $Y(f) = X(f) H(f)$, you can recover $X(f) = Y(f)/H(f)$ (in principle). This is used in echo cancellation, image deblurring, and channel equalization, though noise makes direct division unstable in practice.

Relationship with Laplace transform

The Laplace transform generalizes the CTFT by replacing the real frequency variable with a complex frequency $s = \sigma + j\omega$. Where the CTFT evaluates the signal's response along the imaginary axis only, the Laplace transform explores the entire complex plane, revealing stability and transient behavior.

Similarities and differences

Both transforms share linearity, time shifting, and convolution properties. The key differences:

• The CTFT uses real frequency $f$ (or $\omega = 2\pi f$); the Laplace transform uses complex frequency $s = \sigma + j\omega$
• The CTFT requires the signal to be absolutely integrable (or to exist in a generalized sense with distributions like $\delta$). The Laplace transform can handle growing signals like $e^{at}u(t)$ for $a > 0$, because the $e^{-\sigma t}$ factor in the kernel provides a convergence mechanism.
• The Laplace transform directly encodes stability and causality information through pole locations; the CTFT does not.

Regions of convergence

The region of convergence (ROC) is the set of values of $s$ for which the Laplace integral converges. The ROC determines whether a given pole-zero configuration corresponds to a causal, anti-causal, or two-sided signal. The CTFT exists as a special case of the Laplace transform when the ROC includes the imaginary axis ($\sigma = 0$). If the ROC doesn't include the imaginary axis, the CTFT doesn't converge for that signal.

Stability analysis

A causal LTI system is BIBO stable (bounded input produces bounded output) if and only if all poles of its transfer function $H(s)$ lie in the left half-plane ($\sigma < 0$). Poles in the left half-plane correspond to decaying exponentials in the impulse response. The CTFT alone can't tell you about stability directly, since it only sees the system's behavior along $s = j\omega$. The Laplace transform gives the full picture by showing where the poles sit relative to the imaginary axis.

Sampling and aliasing

Sampling converts a continuous-time signal to a discrete-time signal by recording values at intervals of $T_s = 1/f_s$ seconds. Done correctly, no information is lost. Done incorrectly, aliasing permanently corrupts the signal.

Nyquist sampling theorem

The Nyquist-Shannon sampling theorem states: a band-limited signal with maximum frequency $f_{\max}$ can be perfectly reconstructed from its samples if and only if the sampling rate satisfies:

$f_s > 2f_{\max}$

The critical rate $f_s = 2f_{\max}$ is called the Nyquist rate. For example, audio signals band-limited to 20 kHz require a sampling rate above 40 kHz (CD audio uses 44.1 kHz). Sampling at exactly the Nyquist rate is a theoretical boundary; in practice you always sample somewhat above it.

Aliasing in frequency domain

When you sample at rate $f_s$, the spectrum of the sampled signal consists of copies of $X(f)$ shifted by integer multiples of $f_s$. If $f_s < 2f_{\max}$, these spectral copies overlap, and the overlapping portions add together irreversibly. High-frequency components fold back and masquerade as lower frequencies. Once aliasing occurs, there's no way to separate the overlapped components from the original signal.

Anti-aliasing filters

An anti-aliasing filter is a low-pass filter applied to the continuous-time signal before sampling. Its job is to attenuate all frequency components above $f_s/2$ (the Nyquist frequency) so that the sampled signal satisfies the Nyquist criterion. In practice, ideal brick-wall filters don't exist, so systems use a small guard band: the sampling rate is set somewhat higher than $2f_{\max}$, and a realizable low-pass filter rolls off the spectrum before the Nyquist frequency. Every ADC (analog-to-digital converter) includes an anti-aliasing filter at its input for this reason.