Continuous-time Fourier transform

The continuous-time Fourier transform (CTFT) represents an aperiodic signal as a continuous distribution of frequencies. It is the single most important analytical tool in the course.

Definition (f-form)

The forward transform:

$X(f)=\mathcal{F}\{x(t)\}={\int }_{-\infty }^{\infty }x(t)e^{-j2\pi ft}dt.$

The inverse transform:

$x(t)={\mathcal{F}}^{-1}\{X(f)\}={\int }_{-\infty }^{\infty }X(f)e^{+j2\pi ft}df.$

The two halves differ only in the sign of the exponent. There is no factor of $2\pi$ out front in either direction — that’s the f-form, and that’s what makes it clean.

We write the pair as $x(t)\stackrel{\mathcal{F}}{\leftrightarrow }X(f)$.

ω-form

The ω-form uses radian frequency $\omega =2\pi f$ instead:

$X(j\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt,x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(j\omega )e^{+j\omega t}d\omega .$

The forward direction has no $2\pi$, but the inverse picks up a $1/(2\pi )$ factor — asymmetric. The two forms describe the same mathematical object, related by $X(f)=X(j\omega ){|}_{\omega =2\pi f}$. The course defaults to f-form for clean symmetry; the formula sheet has both.

Why we take T → ∞ of the Fourier series

Start with a Fourier series for a periodic signal of period $T_0$. The harmonic coefficients live at discrete frequencies $k/T_0$, with spacing $1/T_0$. As $T_0\to \infty$, the spacing goes to zero, and the discrete spectrum fills in to become continuous. The sum becomes an integral; the coefficient index $k$ becomes the continuous variable $f$. The result is the CTFT.

This is one of the most beautiful derivations in signals-and-systems, and the punch line is that the CTFT is the Fourier series taken to its aperiodic limit.

What X(f) means

$X(f)$ is the signal’s spectrum:

• Magnitude $|X(f)|$: how much of frequency $f$ is in $x(t)$.
• Phase $\angle X(f)$: where the cosine-sine combination is positioned in time at that frequency.

For real $x(t)$, $|X(f)|$ is even and $\angle X(f)$ is odd (conjugate symmetry).

A signal whose spectrum is concentrated near $f=0$ is lowpass; near a high $f_c$, bandpass; spread out toward high $f$, highpass. See filter note.

Definition memorization

The CTFT forward and inverse definitions (f-form) are not on the formula sheet. They are the third major formula you have to know by heart (after the convolution integral and the Fourier series analysis/synthesis pair). Laplace transform is the fourth.

Standard pairs

The most-used pairs:

• $\delta (t)\leftrightarrow 1$, $1\leftrightarrow \delta (f)$ — duals.
• $\mathrm{rect}(t)\leftrightarrow \mathrm{sinc}(f)$, $\mathrm{sinc}(t)\leftrightarrow \mathrm{rect}(f)$ — duals.
• $\mathrm{tri}(t)\leftrightarrow {\mathrm{sinc}}^2(f)$, ${\mathrm{sinc}}^2(t)\leftrightarrow \mathrm{tri}(f)$ — duals.
• $e^{-\alpha t}u(t)\leftrightarrow 1/(j2\pi f+\alpha )$, $\alpha >0$.
• ${\delta }_{T_0}(t)\leftrightarrow f_0{\delta }_{f_0}(f)$, $f_0=1/T_0$ — the impulse train pair.
• $\cos (2\pi f_{0}t)\leftrightarrow \frac{1}{2}[\delta (f-f_0)+\delta (f+f_0)]$.
• $\sin (2\pi f_{0}t)\leftrightarrow \frac{j}{2}[\delta (f+f_0)-\delta (f-f_0)]$.
• $\mathrm{sgn}(t)\leftrightarrow 1/(j\pi f)$, $u(t)\leftrightarrow \frac{1}{2}\delta (f)+1/(j2\pi f)$.

Reading rect(f/w)

Just like the time-domain rectangle (see Unit rectangle), $\mathrm{rect}(f/w)$ in the frequency domain has width $w$ (edges at $\pm w/2$), not $\pm w$. This is the convention that catches students on every problem involving filtered spectra.

Key properties (all on the formula sheet)

• Linearity: $\alpha x+\beta y\leftrightarrow \alpha X+\beta Y$.
• Time-shifting: $x(t-t_0)\leftrightarrow X(f)e^{-j2\pi f{t}_0}$. Phase rotation linear in $f$; magnitude unchanged.
• Frequency-shifting: $e^{j2\pi f_{0}t}x(t)\leftrightarrow X(f-f_0)$. The dual; foundation of modulation.
• Time-scaling: $x(at)\leftrightarrow (1/|a|)X(f/a)$. The time-bandwidth tradeoff in formula form.
• Time-differentiation: $dx/dt\leftrightarrow j2\pi fX(f)$. Differentiation becomes algebra.
• Convolution–multiplication duality: $x(t)\ast h(t)\leftrightarrow X(f)H(f)$ and $x(t)h(t)\leftrightarrow X(f)\ast H(f)$. See Convolution theorem.
• Parseval: $\int |x{|}^{2}dt=\int |X{|}^{2}df$.
• Total area: $X(0)=\int x(t)dt$ and $x(0)=\int X(f)df$.

What this does for LTI systems

For an LTI system $y=x\ast h$, the convolution theorem gives

$Y(f)=X(f)H(f),$

where $H(f)=\mathcal{F}\{h\}$ is the system’s frequency response. So three routes to the output:

1. Convolve in time.
2. Solve the differential equation.
3. Multiply spectra in frequency and inverse-transform.

Route 3 is usually easiest. It’s why we developed the CTFT.

Generalized transforms

Some signals don’t strictly converge in the Fourier integral (e.g. $x(t)=1$, $\mathrm{sgn}(t)$, $\cos$ over all time). They have generalized transforms — limits of well-behaved approximations — which give the impulse-and-rational-function pairs you see in the pair table. The formal theory is distribution theory; we just use the formulas.