Unit 4: The Continuous Time Fourier Transform and Sampling

ECE220 — Signal And Systems 8 min read

The continuous time Fourier transform (CTFT) extends the Fourier series to signals that are not periodic, mapping a time-domain signal into a continuous distribution of frequency components. Sampling then bridges this analog world with digital processing by asking when discrete samples fully capture a continuous signal. Both rest on the same idea: any well-behaved signal is a superposition of complex exponentials $e^{j\omega t}$ .

Defining conventions used throughout:

Signal notation: $x(t)$ is the time signal; $X(j\omega)$ is its transform; $\omega = 2\pi f$ is angular frequency in rad/s.
Existence: the CTFT exists when $x(t)$ is absolutely integrable, $\int_{-\infty}^{\infty}|x(t)|\,dt < \infty$ , or has finite energy.
Duality of domains: narrow in time implies wide in frequency, and vice versa — a scaling reciprocity that governs every property.
Complex spectrum: $X(j\omega) = |X(j\omega)|e^{j\angle X(j\omega)}$ carries a magnitude spectrum and a phase spectrum.

II. The Continuous Time Fourier Transform

From Fourier series to a continuous spectrum.

A. Representation of aperiodic signals: the continuous time Fourier transform

An aperiodic signal is treated as the limit of a periodic signal whose period $T \to \infty$ , so the discrete Fourier-series lines merge into a continuum.

Derivation idea: for a periodic $\tilde{x}(t)$ with period $T$ , the coefficients are $a_k = \frac{1}{T}\int_{-T/2}^{T/2}x(t)e^{-jk\omega_0 t}\,dt$ . As $T\to\infty$ , spacing $\omega_0 = 2\pi/T \to 0$ and $Ta_k \to X(j\omega)$ .
Analysis equation: converts time to frequency.
TEXT
```
X(jω) = ∫₋∞^∞ x(t) e^(−jωt) dt
```
Synthesis equation: rebuilds the signal from its spectrum.
TEXT
```
x(t) = (1/2π) ∫₋∞^∞ X(jω) e^(jωt) dω
```
Symbol definitions: $X(j\omega)$ is the spectrum (units of amplitude·seconds); the $1/2\pi$ accounts for integrating over $\omega$ rather than $f$ .
Worked example — one-sided exponential: for , :
TEXT
```
X(jω) = ∫₀^∞ e^(−at) e^(−jωt) dt = 1/(a + jω)
```
Magnitude , a lowpass shape decaying with .

B. The Fourier transform for periodic signals

A periodic signal has no finite integral, yet it still has a transform once impulses are allowed.

Key result: a single exponential transforms to an impulse in frequency.
TEXT
```
x(t) = e^(jω₀t)  ⟷  X(jω) = 2π δ(ω − ω₀)
```
General periodic signal: expand as a Fourier series, then transform term by term.
TEXT
```
x(t) = Σ aₖ e^(jkω₀t)  ⟷  X(jω) = 2π Σ aₖ δ(ω − kω₀)
```
Interpretation: the continuous spectrum collapses to a train of impulses located at harmonics $k\omega_0$ , each weighted by $2\pi a_k$ — the CTFT thus subsumes the Fourier series as a special case.
Example: $\cos(\omega_0 t) \Leftrightarrow \pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$ , two impulses symmetric about the origin.

C. Software simulation of frequency spectrum of real world signals

Real signals are sampled and finite, so their spectra are computed numerically with the FFT, an efficient discrete Fourier transform.

Pipeline: acquire samples at rate $f_s$ , window the record, apply the FFT, scale to physical units.
Frequency axis: for $N$ samples, bin $k$ maps to $f_k = k f_s / N$ ; resolution is $\Delta f = f_s/N$ .
Windowing: multiplying by a Hann or Hamming window reduces spectral leakage caused by finite record length.

Illustrative code:

PYTHON

import numpy as np
fs = 1000                       # sampling rate, Hz
t  = np.arange(0, 1, 1/fs)
x  = np.sin(2*np.pi*50*t) + 0.5*np.sin(2*np.pi*120*t)
X  = np.fft.rfft(x * np.hanning(len(x)))
f  = np.fft.rfftfreq(len(x), 1/fs)
mag = np.abs(X) / len(x)

Reading the output: peaks appear near 50 Hz and 120 Hz; the window smears each peak but suppresses side ripples.

D. Properties of continuous time Fourier transform

The properties turn hard integrals into algebra by exploiting how operations in one domain map to the other.

Linearity: $ax_1(t)+bx_2(t) \Leftrightarrow aX_1(j\omega)+bX_2(j\omega)$ .
Time shift: $x(t-t_0) \Leftrightarrow e^{-j\omega t_0}X(j\omega)$ — shifts alter phase only, never magnitude.
Frequency shift (modulation): $e^{j\omega_0 t}x(t) \Leftrightarrow X(j(\omega-\omega_0))$ , the basis of AM communication.
Time scaling: $x(at) \Leftrightarrow \frac{1}{|a|}X(j\omega/a)$ — compressing time stretches frequency.
Convolution: $x(t)*h(t) \Leftrightarrow X(j\omega)H(j\omega)$ , the property that makes frequency-domain filtering possible.
Multiplication: $x(t)h(t) \Leftrightarrow \frac{1}{2\pi}X(j\omega)*H(j\omega)$ , the dual of convolution.
Differentiation: $\frac{d}{dt}x(t) \Leftrightarrow j\omega X(j\omega)$ ; each derivative multiplies by $j\omega$ .

Parseval's relation: energy is conserved across domains.

TEXT

∫₋∞^∞ |x(t)|² dt = (1/2π) ∫₋∞^∞ |X(jω)|² dω

III. Sampling

Turning a continuous signal into samples without losing information.

A. Sampling: Introduction

Sampling records a continuous signal only at instants spaced $T$ apart, producing a sequence that a digital system can store and process.

Ideal sampling model: multiply by an impulse train (the sampling function).
TEXT
```
p(t) = Σ δ(t − nT)      xₚ(t) = x(t) p(t) = Σ x(nT) δ(t − nT)
```
Sampling parameters: $T$ is the sampling period; $f_s = 1/T$ is the sampling rate; $\omega_s = 2\pi/T$ .
Central question: whether the discrete samples $x(nT)$ preserve enough information to recover $x(t)$ exactly.

B. Representation of continuous time signal by its samples: sampling theorem

The sampling theorem states the exact condition under which samples fully determine a bandlimited signal.

Statement: a signal bandlimited to (i.e. for ) is uniquely determined by samples spaced apart provided
TEXT
```
ωₛ > 2ωM      equivalently      fs > 2 fM
```
Nyquist rate: the minimum rate $2f_M$ ; $f_s/2$ is the Nyquist frequency, the highest recoverable component.
Spectral picture: sampling in time replicates the spectrum every .
TEXT
```
Xₚ(jω) = (1/T) Σ X(j(ω − kωₛ))
```
Why the bound works: if $\omega_s > 2\omega_M$ the replicas do not overlap, so a lowpass filter can isolate the original baseband copy.
Example: audio bandlimited to 20 kHz needs $f_s > 40$ kHz; the CD standard 44.1 kHz satisfies this with margin.

C. Reconstruction of a signal from its samples using interpolation

Reconstruction fills the gaps between samples by lowpass filtering the impulse-weighted samples.

Ideal reconstruction filter: an ideal lowpass with cutoff $\omega_c$ between $\omega_M$ and $\omega_s-\omega_M$ and gain $T$ ; its impulse response is a sinc.

Sinc interpolation formula: