Signals, LTI Systems & the Frequency Domain

1. signal
2. system
3. convolution
4. spectrum
5. wave

Each idea is the tool needed to understand the next one. Nothing here is optional background: the whole of amplitude modulation in Weeks 2–4 is built from these five.

A signal carries information through variation. A system receives a signal and produces another. For a particularly useful class — linear time-invariant systems — the impulse response tells us how every input will be transformed. Convolution connects the input and impulse response to the output.

Part 1

Signals

Section 01 — making variation visible, and the three numbers that describe a sinusoid.

A continuous-time sinusoid can be written as

Amplitude \(A\) sets vertical scale, frequency \(f\) counts cycles per second, and phase \(\phi\) sets the waveform's position within a cycle at the reference time.

Frequency in hertz is not angular frequency: \(\omega=2\pi f\) has units of radians per second.

Part 2

LTI systems and convolution

Sections 02–03 — two promises that make a system tractable, and the operation that follows from them.

Linearity means scaled and added inputs produce correspondingly scaled and added outputs.

Time invariance means delaying the input delays the output by the same amount without changing its shape.

The response of an LTI system to \(\delta(t)\) is its impulse response \(h(t)\). Because signals can be represented as weighted, shifted impulses, linearity and time invariance let us construct the response to any suitable input from shifted copies of \(h(t)\).

One integral, evaluated once for every output time \(t\).

For each output time \(t\), begin with \(h(\tau)\), reverse it to obtain \(h(-\tau)\), and shift it to obtain \(h(t-\tau)\). Compare it with \(x(\tau)\), multiply where the two signals overlap, and integrate over \(\tau\). That area gives one value \(y(t)\); repeat the process for other values of \(t\).

For discrete-time signals, the corresponding operation is a summation rather than an integral.

1. Reverse \(h(\tau)\) to obtain \(h(-\tau)\), then shift by \(t\).
2. Multiply \(h(t-\tau)\) with \(x(\tau)\) wherever they overlap.
3. Integrate over \(\tau\) to obtain the current output value \(y(t)\).

Part 3

The frequency domain

Section 04 — the same signal, seen from a second viewpoint.

The time domain shows how a signal changes with time. The frequency domain shows which sinusoidal frequencies contribute and with what relative magnitude. A pure real cosine produces components at positive and negative frequency in a two-sided spectrum.

Part 4

Waves in space

Section 05 — from a frequency measured in time to a length measured in metres.

At a fixed location, frequency \(f\) measures cycles per second. At a fixed time, wavelength \(\lambda\) measures the distance between equal-phase points. Phase velocity \(v_p\) connects the two:

Use hertz for \(f\), metres per second for \(v_p\), and metres for \(\lambda\).

In the explorer, frequency is entered in megahertz and converted internally to hertz.

Part 5

MATLAB, and the week in four relationships

Sections 06–08 — give the ideas a computational form, then check that you can connect the representations.

This example creates a sampled sinusoid and a correctly scaled one-sided magnitude spectrum. The FFT is the numerical algorithm used to compute samples of the discrete Fourier transform.

% Week 1: time-domain sinusoid and magnitude spectrum
Fs = 1000;                    % sampling frequency in hertz
T  = 1/Fs;                   % sampling interval
L  = 1000;                   % number of samples (1 second)
t  = (0:L-1)*T;              % time vector: 0 to (L-1)/Fs

A   = 1.0;                   % amplitude
f0  = 50;                    % sinusoidal frequency in hertz
phi = 0;                     % phase in radians
x = A*cos(2*pi*f0*t + phi);

figure;
plot(t, x, 'LineWidth', 1.4);
xlim([0 0.1]); grid on;
xlabel('Time (s)'); ylabel('Amplitude');
title('Time-domain signal');

% Compute a one-sided magnitude spectrum
X = fft(x);
P2 = abs(X/L);                % normalized two-sided magnitude
P1 = P2(1:floor(L/2)+1);      % retain nonnegative frequencies
P1(2:end-1) = 2*P1(2:end-1); % account for removed negative half
f = Fs*(0:floor(L/2))/L;      % frequency axis in hertz

figure;
stem(f, P1, 'filled');
xlim([0 150]); grid on;
xlabel('Frequency (Hz)'); ylabel('Magnitude');
title('One-sided magnitude spectrum');

% Try these one at a time, then recreate x and rerun the plots:
% A = 2.0;        % doubles time amplitude and spectral magnitude
% f0 = 100;       % doubles oscillation rate; peak moves to 100 Hz
% phi = pi/2;     % shifts the waveform; ideal magnitude stays unchanged

The bin spacing here is \(F_s/L=1\,\text{Hz}\), so \(50\,\text{Hz}\) and \(100\,\text{Hz}\) fall exactly on FFT bins and leakage is avoided in these examples.

EE 351 · Week 1

From representation to a system that carries information

MATLAB gives these ideas a computational form: construct a time axis, generate the signal, plot it, compute an FFT, build the matching frequency axis, and interpret what changed — not merely what the software displayed.

Continue to Week 2 — Communication Systems →