Properties: Convolution, Modulation and Energy

Shifts, scaling and modulation, the convolution theorem, Parseval's relation and the time–bandwidth limit

London — 1946

Dennis Gabor, a Hungarian-born engineer working in Britain, publishes "Theory of Communication" in the Journal of the Institution of Electrical Engineers (Gabor transform). He asks a question about telephone and radio channels: how much information can a signal of a given duration and bandwidth carry? On the way he proves that no signal can be sharply confined both in time and in frequency at once — the product of its duration and its bandwidth has a floor. He proposes describing signals with Gaussian-shaped elementary pulses that sit exactly on that floor. Gabor later received the 1971 Nobel Prize in Physics for inventing holography (Dennis Gabor). The floor he found in 1946 is now called the Gabor limit — the uncertainty principle of signal processing. This lesson collects the transform's properties, of which the time–bandwidth limit is the deepest, and the convolution theorem the most useful.

💡The properties

| In time | In frequency | |---|---| |

a\,x(t) + b\,y(t)

a\,X(j\omega) + b\,Y(j\omega)

| |

x(t - t_0)

e^{-j\omega t_0}X(j\omega)

| |

e^{j\omega_0 t}x(t)

X\big(j(\omega - \omega_0)\big)

| |

x(at)

\frac{1}{|a|}X\!\left(\frac{j\omega}{a}\right)

| |

dx/dt

j\omega\,X(j\omega)

| |

(x * h)(t)

X(j\omega)\,H(j\omega)

| |

x(t)\,y(t)

\frac{1}{2\pi}(X * Y)(j\omega)

| |

\int |x|^2\,dt

\frac{1}{2\pi}\int |X|^2\,d\omega

| Read the table both ways: a delay is a phase ramp; compressing time stretches frequency; multiplying by a carrier slides the spectrum; and — the reason the transform rules engineering — convolution becomes multiplication.

Derivation

The convolution theorem

Transform

y(t) = \int x(\tau)\,h(t - \tau)\,d\tau

and swap the order of integration:

Y(j\omega) = \int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty} x(\tau)\,h(t - \tau)\,d\tau\right] e^{-j\omega t}\,dt = \int_{-\infty}^{\infty} x(\tau)\left[\int_{-\infty}^{\infty} h(t - \tau)\,e^{-j\omega t}\,dt\right] d\tau .

In the inner integral put

u = t - \tau

: it becomes

e^{-j\omega\tau}\int h(u)\,e^{-j\omega u}\,du = e^{-j\omega\tau}H(j\omega)

. Then

Y(j\omega) = H(j\omega)\int_{-\infty}^{\infty} x(\tau)\,e^{-j\omega\tau}\,d\tau = H(j\omega)\,X(j\omega).

The output spectrum is the input spectrum times the frequency response. Every LTI system is completely described by

H(j\omega)

, and cascading systems multiplies their responses.

Derivation

Delay and scaling

Delay. Substitute

u = t - t_0

\int x(t - t_0)\,e^{-j\omega t}\,dt = \int x(u)\,e^{-j\omega(u + t_0)}\,du = e^{-j\omega t_0}X(j\omega).

A delay leaves the magnitude alone and adds a phase that grows linearly with frequency,

-\omega t_0

. Hence a filter whose phase is linear in frequency delays every component equally — it does not distort.

Scaling. For

a > 0

substitute

u = at

\int x(at)\,e^{-j\omega t}\,dt = \frac{1}{a}\int x(u)\,e^{-j(\omega/a)u}\,du = \frac1a X\!\left(\frac{j\omega}{a}\right).

Compressing by

a

stretches the spectrum by

a

— the phonautogram played at double speed (Lesson 1-1) had every frequency doubled.

📐Modulation: moving a spectrum

Multiply a message

m(t)

by a carrier

\cos\omega_c t = \tfrac12(e^{j\omega_c t} + e^{-j\omega_c t})

. By the frequency-shift property,

m(t)\cos\omega_c t \;\leftrightarrow\; \tfrac12 M\big(j(\omega - \omega_c)\big) + \tfrac12 M\big(j(\omega + \omega_c)\big).

The message's spectrum is copied, at half height, to

\pm\omega_c

. A voice band of 0–4 kHz on a 1 MHz carrier occupies 996–1004 kHz. Multiply by the carrier again and the copies come back to zero frequency (plus copies at

\pm2\omega_c

, which a low-pass filter removes). That is how every radio transmitter and receiver moves signals — Unit 10.

💡Parseval's relation for transforms

\int_{-\infty}^{\infty} |x(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(j\omega)|^2\,d\omega .

Energy can be counted in time or in frequency.

|X(j\omega)|^2

is the energy spectral density: how a signal's energy is distributed over frequency. For a filter,

|H|^2

says how much of the input's energy at each frequency gets through.

The time–bandwidth limit. Measure a signal's spread in time by its rms duration

\Delta t

(the standard deviation of

|x(t)|^2

treated as a distribution) and its spread in frequency by its rms bandwidth

\Delta\omega

(the same for

|X(j\omega)|^2

). Then

\Delta t\,\Delta\omega \;\ge\; \tfrac12 ,

with equality only for Gaussian pulses. A short signal must be wideband; a narrowband signal must last long. This is not a limitation of instruments but of signals themselves — the same mathematics as Heisenberg's uncertainty principle, which is why it bears Gabor's name in engineering.

📐The Gaussian sits on the limit

Take

x(t) = e^{-t^2/2\sigma^2}

. Then

|x|^2 = e^{-t^2/\sigma^2}

, a Gaussian with standard deviation

\sigma/\sqrt2

, so

\Delta t = \sigma/\sqrt2

. Its transform

X \propto e^{-\sigma^2\omega^2/2}

gives

|X|^2 \propto e^{-\sigma^2\omega^2}

, with standard deviation

\Delta\omega = 1/(\sigma\sqrt2)

. The product is

\Delta t\,\Delta\omega = \frac{\sigma}{\sqrt2}\cdot\frac{1}{\sigma\sqrt2} = \frac12,

exactly the floor, whatever

\sigma

. A rectangular pulse does worse: its sharp edges spray energy far out in frequency, and its rms bandwidth is in fact infinite.

Incomplete knowledge of the future, and also of the past of the transmitter from which the future might be constructed, is at the very basis of the concept of information.

– Dennis Gabor, "Optical transmission", in Information Theory: Papers Read at a Symposium on Information Theory (1952), quoted on Wikiquote

⚠Multiplying in time smears in frequency

The table runs both ways. Cutting a recording to a finite window multiplies it by a rectangle, so its spectrum is convolved with a sinc: every sharp spectral line spreads out into a main lobe and side lobes. That is spectral leakage, the subject of Lesson 6-3, and the reason analysers use smooth windows.

💡Why radar wants short pulses — and long ones

Range resolution needs a short pulse (two echoes closer than the pulse length blur together); detecting a faint echo needs energy, which a short pulse at limited peak power does not have. The time–bandwidth trade seems to forbid both. Radar escapes it with pulse compression: a long pulse whose frequency sweeps, so it has a large bandwidth anyway, compressed on reception by a matched filter (Lesson 10-3).

💡Where you will use this

Multiplication by

H(j\omega)

is the whole of Lessons 4-3 and 4-4. Modulation is Unit 10; leakage and windows are Lesson 6-3; the time–frequency trade-off is the spectrogram's dilemma in Lesson 6-4. In discrete time the same table holds for the DTFT (Lesson 6-1).

📚Further reading

Oppenheim and Willsky, Signals and Systems, sections 4.3–4.6 — properties of the continuous-time Fourier transform, the convolution and multiplication properties.
Dennis Gabor, "Theory of Communication", Journal of the IEE, Part III, 93 (1946), 429–457 — the original time–frequency paper.
Gabor transform on Wikipedia.

↓But there is a problem

The convolution theorem says an LTI system is nothing more than its frequency response:

Y = HX

. So everything about a filter, an amplifier, a microphone or a feedback loop is in one complex function of frequency. How do engineers see that function — and sketch it in a minute, with a ruler, for a system they have only just written down? In the 1930s a Bell Labs mathematician found that on the right axes most of it is straight lines.