Part I — Continuous-Domain Foundations

Chapter 3: The Fourier Transform

The Fourier series decomposes periodic signals into discrete harmonics. When we let the period grow to infinity, the discrete sum becomes an integral over a continuous spectrum: the Fourier transform. This chapter develops the theory, key properties, the convolution theorem, Parseval's relation, the uncertainty principle, and the short-time Fourier transform for time-frequency analysis.

3.1 From Series to Transform

Recall from Chapter 2 that a periodic signal with fundamental period $T_0$ and fundamental frequency $f_0 = 1/T_0$ admits the Fourier series representation:

$$x(t) = \sum_{n=-\infty}^{\infty} c_n\, e^{j2\pi n f_0 t}, \qquad c_n = \frac{1}{T_0}\int_{-T_0/2}^{T_0/2} x(t)\,e^{-j2\pi n f_0 t}\,dt$$

To handle aperiodic signals, we consider the limiting case $T_0 \to \infty$. Define $f_n = n f_0 = n/T_0$ and $\Delta f = f_0 = 1/T_0$. The coefficients become

$$c_n = \frac{1}{T_0}\int_{-T_0/2}^{T_0/2} x(t)\,e^{-j2\pi f_n t}\,dt = \frac{\Delta f}{1}\int_{-T_0/2}^{T_0/2} x(t)\,e^{-j2\pi f_n t}\,dt$$

Define the spectral envelope:

$$X(f_n) \;=\; \int_{-T_0/2}^{T_0/2} x(t)\,e^{-j2\pi f_n t}\,dt$$

so that $c_n = \Delta f \cdot X(f_n)$. Substituting back into the synthesis equation:

$$x(t) = \sum_{n=-\infty}^{\infty} X(f_n)\,e^{j2\pi f_n t}\,\Delta f$$

As $T_0 \to \infty$, we have $\Delta f \to 0$, the discrete frequencies $f_n$ fill the real line, and the Riemann sum becomes an integral:

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

Key intuition: The Fourier series decomposes a periodic signal into a discrete set of harmonics spaced by $f_0$. As the period grows, the harmonics move closer together, and in the limit they merge into a continuous spectral density $X(f)$.

3.2 The Continuous Fourier Transform

Definition — Fourier Transform (Forward)

Given a signal $x(t)$, its Fourier transform is the function$X(f)$ defined by:

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

We also write $x(t) \xleftrightarrow{\mathcal{F}} X(f)$ to denote a transform pair.

Definition — Inverse Fourier Transform

The original signal is recovered from its spectrum by the inverse transform:

$$x(t) = \mathcal{F}^{-1}\{X(f)\} = \int_{-\infty}^{\infty} X(f)\,e^{j2\pi f t}\,df$$

Angular frequency convention: Many textbooks use $\omega = 2\pi f$, giving$X(\omega) = \int x(t)\,e^{-j\omega t}\,dt$ and$x(t) = \frac{1}{2\pi}\int X(\omega)\,e^{j\omega t}\,d\omega$. The factor $1/(2\pi)$ shifts between forward and inverse transforms depending on the convention. This course primarily uses the frequency-in-Hz convention ($f$), which places factors symmetrically.

Existence Conditions

The Fourier transform integral converges absolutely when $x(t)$ is absolutely integrable:

$$\int_{-\infty}^{\infty} |x(t)|\,dt < \infty \qquad (x \in L^1(\mathbb{R}))$$

Under this condition, $X(f)$ is continuous, bounded, and$|X(f)| \leq \int |x(t)|\,dt$ for all $f$. The Riemann-Lebesgue lemma further guarantees $X(f) \to 0$ as$|f| \to \infty$.

More generally, the Fourier transform extends to $L^2(\mathbb{R})$(square-integrable functions) via the Plancherel theorem (Section 3.5), and even to tempered distributions (allowing us to define the transform of $\delta(t)$, constants, and sinusoids).

Theorem — Riemann-Lebesgue Lemma

If $x \in L^1(\mathbb{R})$, then $X(f) = \mathcal{F}\{x\}(f)$ is continuous and satisfies $\lim_{|f|\to\infty} X(f) = 0$.

3.3 Key Properties

The power of the Fourier transform lies in its rich algebraic properties. Operations in the time domain correspond to simple operations in the frequency domain and vice versa. Let $x(t) \xleftrightarrow{\mathcal{F}} X(f)$ and$y(t) \xleftrightarrow{\mathcal{F}} Y(f)$.

Property	Time Domain	Frequency Domain
Linearity	$a\,x(t) + b\,y(t)$	$a\,X(f) + b\,Y(f)$
Time Shift	$x(t - t_0)$	$e^{-j2\pi f t_0}\,X(f)$
Frequency Shift	$e^{j2\pi f_0 t}\,x(t)$	$X(f - f_0)$
Scaling	$x(at)$	$\frac{1}{\|a\|}\,X\!\left(\frac{f}{a}\right)$
Time Reversal	$x(-t)$	$X(-f)$
Convolution	$x(t) * y(t)$	$X(f)\,Y(f)$
Multiplication	$x(t)\,y(t)$	$X(f) * Y(f)$
Differentiation	$\frac{d^n x}{dt^n}$	$(j2\pi f)^n\,X(f)$
Integration	$\int_{-\infty}^{t} x(\tau)\,d\tau$	$\frac{X(f)}{j2\pi f} + \frac{X(0)}{2}\,\delta(f)$
Duality	$X(t)$	$x(-f)$

Example — Time-Shift Property Proof

Let $y(t) = x(t - t_0)$. Then:

$$Y(f) = \int_{-\infty}^{\infty} x(t-t_0)\,e^{-j2\pi ft}\,dt$$

Substitute $\tau = t - t_0$, so $t = \tau + t_0$ and $dt = d\tau$:

$$Y(f) = \int_{-\infty}^{\infty} x(\tau)\,e^{-j2\pi f(\tau + t_0)}\,d\tau = e^{-j2\pi f t_0}\int_{-\infty}^{\infty} x(\tau)\,e^{-j2\pi f\tau}\,d\tau = e^{-j2\pi f t_0}\,X(f)$$

A time delay of $t_0$ multiplies the spectrum by a linear phase factor. The magnitude spectrum is unchanged: $|Y(f)| = |X(f)|$.

Example — Differentiation Property

Starting from the inverse transform $x(t) = \int X(f)\,e^{j2\pi ft}\,df$, differentiate both sides:

$$\frac{dx}{dt} = \int X(f)\cdot j2\pi f\,e^{j2\pi ft}\,df = \mathcal{F}^{-1}\{j2\pi f\,X(f)\}$$

Hence $\mathcal{F}\left\{\frac{dx}{dt}\right\} = j2\pi f\,X(f)$. By induction, the $n$-th derivative gives the factor $(j2\pi f)^n$. This converts differential equations into algebraic equations — the key idea behind the Laplace and Fourier transform methods for solving ODEs/PDEs.

Duality: If you know$x(t) \leftrightarrow X(f)$, then you automatically know$X(t) \leftrightarrow x(-f)$. This is extremely powerful: every transform pair immediately gives a second one for free.

3.4 The Convolution Theorem

The convolution theorem is arguably the single most important result in signal processing. It states that convolution in one domain corresponds to multiplication in the other.

Theorem — Convolution Theorem

If $x(t) \xleftrightarrow{\mathcal{F}} X(f)$ and $y(t) \xleftrightarrow{\mathcal{F}} Y(f)$, then:

$$\mathcal{F}\{x * y\}(f) = X(f)\cdot Y(f)$$

where the convolution is defined as:

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

Proof Sketch

Start with the definition:

$$\mathcal{F}\{x*y\}(f) = \int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty} x(\tau)\,y(t-\tau)\,d\tau\right] e^{-j2\pi ft}\,dt$$

Swap the order of integration (justified by Fubini's theorem for$L^1$ functions):

$$= \int_{-\infty}^{\infty} x(\tau) \left[\int_{-\infty}^{\infty} y(t-\tau)\,e^{-j2\pi ft}\,dt\right] d\tau$$

The inner integral is $e^{-j2\pi f\tau}Y(f)$ by the time-shift property. Hence:

$$= \int_{-\infty}^{\infty} x(\tau)\,e^{-j2\pi f\tau}\,d\tau \;\cdot\; Y(f) = X(f)\,Y(f) \qquad \blacksquare$$

Multiplication-Convolution Duality

By the duality of the Fourier transform, there is a companion result:

Theorem — Multiplication Theorem

$$\mathcal{F}\{x(t)\cdot y(t)\} = X(f) * Y(f) = \int_{-\infty}^{\infty} X(\nu)\,Y(f-\nu)\,d\nu$$

Multiplication in time corresponds to convolution in frequency. This explains, for instance, why windowing a signal (multiplying by a window function) causes spectral smearing (convolution with the window's spectrum).

Practical consequence: LTI system analysis becomes trivially easy: the output spectrum is $Y(f) = H(f)\,X(f)$, where $H(f)$ is the system's frequency response. This converts convolution integrals into pointwise multiplication.

3.5 Plancherel / Parseval Theorem (Continuous)

One of the most beautiful results in Fourier analysis is that the transform preserves energy. The total energy computed in the time domain equals the total energy computed in the frequency domain.

Theorem — Parseval's Theorem (Energy Conservation)

If $x(t) \xleftrightarrow{\mathcal{F}} X(f)$, then:

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

The quantity $|X(f)|^2$ is called the **energy spectral density** (ESD). It describes how the signal's energy is distributed across frequencies.

Theorem — Plancherel Theorem (Generalized)

More generally, for $x, y \in L^2(\mathbb{R})$:

$$\int_{-\infty}^{\infty} x(t)\,\overline{y(t)}\,dt = \int_{-\infty}^{\infty} X(f)\,\overline{Y(f)}\,df$$

In other words, the Fourier transform is a **unitary operator** on $L^2(\mathbb{R})$. It preserves the inner product, and hence all geometric notions (angles, distances, orthogonality).

Example — Energy of a Rectangular Pulse

Let $x(t) = \mathrm{rect}(t/T) = \begin{cases} 1, & |t| \leq T/2 \\ 0, & \text{otherwise}\end{cases}$. Then:

$$E_{\text{time}} = \int_{-T/2}^{T/2} 1^2\,dt = T$$

The Fourier transform is $X(f) = T\,\mathrm{sinc}(fT)$. By Parseval's:

$$E_{\text{freq}} = \int_{-\infty}^{\infty} T^2\,\mathrm{sinc}^2(fT)\,df = T$$

This confirms $\int \mathrm{sinc}^2(u)\,du = 1$, a result that can be tricky to prove directly.

Physical interpretation: Parseval's theorem says that a signal's total energy is the same whether you measure it by integrating the squared amplitude over time or over frequency. You cannot create or destroy energy by taking a Fourier transform — it merely redistributes the representation.

3.6 Heisenberg Uncertainty Principle

The Fourier transform imposes a fundamental trade-off: a signal cannot be simultaneously concentrated in both time and frequency. This is the uncertainty principle, which arises purely from the mathematics of Fourier analysis and has profound implications in quantum mechanics, signal processing, and information theory.

Definition — Time-Frequency Spread

Define the **time spread** (standard deviation in time) and **frequency spread** of a signal $x(t)$ with Fourier transform $X(f)$:

$$\sigma_t^2 = \frac{\int t^2\,|x(t)|^2\,dt}{\int |x(t)|^2\,dt}, \qquad \sigma_f^2 = \frac{\int f^2\,|X(f)|^2\,df}{\int |X(f)|^2\,df}$$

(assuming $x$ is centered so that $\int t\,|x(t)|^2\,dt = 0$ and $\int f\,|X(f)|^2\,df = 0$).

Theorem — Gabor-Heisenberg Uncertainty Principle

For any signal $x(t) \in L^2(\mathbb{R})$ with $tx(t) \in L^2(\mathbb{R})$:

$$\sigma_t \cdot \sigma_f \;\geq\; \frac{1}{4\pi}$$

In angular frequency ($\omega = 2\pi f$), this becomes $\sigma_t \cdot \sigma_\omega \geq \frac{1}{2}$.

**Equality** is achieved if and only if $x(t)$ is a Gaussian: $x(t) = Ce^{-\alpha t^2}$ for some constants $C$ and $\alpha > 0$. The Gaussian is the **minimum-uncertainty signal**.

Proof Outline

The proof uses the Cauchy-Schwarz inequality. Consider the inner product:

$$\left|\int t\,x(t)\,\overline{x'(t)}\,dt\right|^2 \leq \left(\int t^2|x(t)|^2\,dt\right)\left(\int |x'(t)|^2\,dt\right)$$

Integration by parts on the left side yields $-\frac{1}{2}\|x\|^2$, and by Parseval's theorem $\int |x'(t)|^2\,dt = (2\pi)^2\int f^2|X(f)|^2\,df$. Combining gives the desired inequality. Equality in Cauchy-Schwarz requires$x'(t) = -2\alpha t\,x(t)$, whose solution is the Gaussian.

Connection to Quantum Mechanics

In quantum mechanics, the wave function $\psi(x)$ and its momentum-space representation $\tilde{\psi}(p)$ are related by a Fourier transform (with $\hbar$ as the conversion factor). The uncertainty principle becomes:

$$\Delta x \cdot \Delta p \;\geq\; \frac{\hbar}{2}$$

This is not a statement about measurement limitations but a fundamental property of Fourier conjugate variables. The same mathematics governs position-momentum in physics, time-frequency in signal processing, and many other dual pairs.

Engineering consequence: You cannot design a filter that is arbitrarily narrow in both time and frequency. A short pulse must have a wide bandwidth. A narrowband signal must be long in duration. This trade-off is inescapable and motivates wavelet and time-frequency analysis methods.

3.7 Common Transform Pairs

The following table collects the most important Fourier transform pairs. These should be committed to memory, as they appear constantly in practice.

Signal Name	Time Domain $x(t)$	Frequency Domain $X(f)$
Rectangular pulse	$\mathrm{rect}(t/T)$	$T\,\mathrm{sinc}(fT)$
Sinc function	$\mathrm{sinc}(Wt)$	$\frac{1}{W}\,\mathrm{rect}(f/W)$
Gaussian	$e^{-\pi t^2}$	$e^{-\pi f^2}$
Two-sided exponential	$e^{-a\|t\|},\; a>0$	$\frac{2a}{a^2 + (2\pi f)^2}$
Dirac delta	$\delta(t)$	$1$
Constant	$1$	$\delta(f)$
Cosine	$\cos(2\pi f_0 t)$	$\frac{1}{2}[\delta(f-f_0)+\delta(f+f_0)]$
Sine	$\sin(2\pi f_0 t)$	$\frac{1}{2j}[\delta(f-f_0)-\delta(f+f_0)]$
Signum	$\mathrm{sgn}(t)$	$\frac{1}{j\pi f}$
Unit step	$u(t)$	$\frac{1}{2}\delta(f) + \frac{1}{j2\pi f}$
Comb (Shah)	$\sum_{n}\delta(t - nT)$	$\frac{1}{T}\sum_{k}\delta(f - k/T)$

Example — Gaussian Self-Reciprocity

The Gaussian $g(t) = e^{-\pi t^2}$ transforms to $G(f) = e^{-\pi f^2}$ — it is its own Fourier transform! This is a unique property. No other function (up to scaling) has this self-reciprocity. The proof proceeds by showing $g(t)$ satisfies the ODE $g'(t) = -2\pi t\,g(t)$, taking the Fourier transform of both sides, and using the differentiation property to obtain the same ODE for $G(f)$.

Duality check: Note how the rect-sinc pair appears twice in the table (once directly and once via duality). Similarly, the delta-constant pair exhibits $\delta(t) \leftrightarrow 1$ and$1 \leftrightarrow \delta(f)$. Every row in the table generates a second row by the duality property.

3.8 Short-Time Fourier Transform (STFT)

The standard Fourier transform provides complete frequency information but discards all time localization: $X(f)$ tells us which frequencies are present but not when they occur. For signals with time-varying spectral content (speech, music, radar returns, seismic data), we need a time-frequency representation.

Definition — Short-Time Fourier Transform

The STFT of a signal $x(t)$ with respect to a window function $w(t)$ is:

$$\mathrm{STFT}\{x\}(\tau, f) = \int_{-\infty}^{\infty} x(t)\,w(t-\tau)\,e^{-j2\pi ft}\,dt$$

Here $\tau$ is the time center of the window, and $f$ is frequency. The result is a **two-dimensional** function of time and frequency.

The idea is simple: slide a short window $w(t-\tau)$ along the signal, and at each position compute the Fourier transform of the windowed segment. The squared magnitude $|\mathrm{STFT}(\tau,f)|^2$ is the spectrogram, which shows how spectral energy varies over time.

Time-Frequency Resolution Trade-off

The uncertainty principle directly constrains the STFT:

A short window gives good time resolution (events are localized in time) but poor frequency resolution (wide main lobe in the window's spectrum).
A long window gives good frequency resolution but poor time resolution — transient events get smeared.

The STFT tiles the time-frequency plane with rectangles of fixed size$\Delta t \times \Delta f$, where $\Delta t \cdot \Delta f \geq \frac{1}{4\pi}$. Once you choose the window, the tile size is fixed everywhere. This is a fundamental limitation of the STFT, and it motivates the wavelet transform (Chapter 9), which uses tiles that adapt their aspect ratio to frequency.

Definition — Spectrogram

The **spectrogram** is the squared magnitude of the STFT:

$$S(\tau, f) = |\mathrm{STFT}\{x\}(\tau, f)|^2$$

It is commonly displayed as a heat map with time on the horizontal axis, frequency on the vertical axis, and color/brightness representing power.

Common Window Functions

The choice of window affects the trade-off:

Rectangular: Narrowest main lobe but highest sidelobes (-13 dB). Best frequency resolution for a given length, worst spectral leakage.
Hann/Hamming: Good general-purpose choice. Sidelobes at -31 dB (Hann) or -43 dB (Hamming).
Gaussian: Achieves the theoretical minimum uncertainty product. Optimal in the uncertainty-principle sense.
Kaiser: Parameterized window that allows continuous tuning between narrow main lobe and low sidelobes.

Beyond the STFT: The fixed resolution of the STFT motivates more advanced time-frequency representations: the wavelet transform (multi-resolution), Wigner-Ville distribution(bilinear, optimal resolution but cross-term artifacts), and reassigned spectrograms (sharpened localization). These are explored in later chapters.

Interactive Python Examples

The following interactive examples let you compute Fourier transforms numerically and visualize the key concepts from this chapter.

Fourier Transform of Common Signals

We compute the FFT of four canonical signals — rectangular pulse, Gaussian, exponential decay, and windowed cosine — and display their time-domain waveforms alongside their magnitude spectra. Compare the results with the analytical transform pairs in the table above.

import numpy as np
import matplotlib.pyplot as plt

N = 4096
fs = 100
t = np.arange(N) / fs - N/(2*fs)
freq = np.fft.fftshift(np.fft.fftfreq(N, 1/fs))

signals = {
    'Rectangular Pulse': np.where(np.abs(t) <= 1, 1.0, 0.0),
    'Gaussian': np.exp(-t**2 / (2*0.5**2)),
    'Exponential Decay': np.exp(-2*np.abs(t)),
    'Cosine (5 Hz)': np.cos(2*np.pi*5*t) * np.where(np.abs(t) <= 3, 1.0, 0.0),
}

fig, axes = plt.subplots(4, 2, figsize=(12, 12))
fig.suptitle('Fourier Transform Pairs', color='white', fontsize=14)

for i, (name, sig) in enumerate(signals.items()):
    spectrum = np.fft.fftshift(np.fft.fft(sig)) / fs

axes[i, 0].plot(t, sig, color='#818cf8', linewidth=2)
    axes[i, 0].set_title(f'{name} — Time', color='#a5b4fc', fontsize=11)
    axes[i, 0].set_xlim(-5, 5)

axes[i, 1].plot(freq, np.abs(spectrum), color='#c084fc', linewidth=2)
    axes[i, 1].set_title(f'{name} — |X(f)|', color='#a5b4fc', fontsize=11)
    axes[i, 1].set_xlim(-15, 15)

for ax in axes.flat:
    ax.grid(True, alpha=0.3)
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')

plt.tight_layout()
plt.show()

Click Run to execute the Python code

First run will download Python environment (~15MB)

Uncertainty Principle Demonstration

This example demonstrates the Heisenberg uncertainty principle using Gaussian pulses of varying width. A narrow Gaussian in time ($\sigma_t = 0.1$) produces a wide spectrum, while a broad Gaussian ($\sigma_t = 2.0$) produces a narrow spectrum. The product $\sigma_t \cdot \sigma_\omega$ remains constant at$1/2$ — the theoretical minimum.

import numpy as np
import matplotlib.pyplot as plt

N = 4096
fs = 100
t = np.arange(N) / fs - N/(2*fs)
freq = np.fft.fftshift(np.fft.fftfreq(N, 1/fs))

fig, axes = plt.subplots(3, 2, figsize=(12, 9))
fig.suptitle('Heisenberg Uncertainty: Narrow in time <-> Wide in frequency', color='white', fontsize=14)

sigmas = [0.1, 0.5, 2.0]
for i, sigma in enumerate(sigmas):
    g = np.exp(-t**2 / (2*sigma**2))
    G = np.fft.fftshift(np.fft.fft(g)) / fs

sigma_t = sigma
    sigma_f = 1 / (2*np.pi*sigma)
    product = sigma_t * (2*np.pi*sigma_f)

axes[i,0].plot(t, g, color='#818cf8', linewidth=2)
    axes[i,0].set_title(f'sigma_t = {sigma_t:.1f}', color='#a5b4fc')
    axes[i,0].set_xlim(-5, 5)

axes[i,1].plot(freq, np.abs(G), color='#c084fc', linewidth=2)
    axes[i,1].set_title(f'sigma_omega = {sigma_f:.2f},  sigma_t * sigma_omega = {product:.2f}', color='#a5b4fc')
    axes[i,1].set_xlim(-5, 5)

for ax in axes.flat:
    ax.grid(True, alpha=0.3)
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')

plt.tight_layout()
plt.show()

print("For Gaussian pulses, sigma_t * sigma_omega = 1/2 (minimum uncertainty)")

Click Run to execute the Python code

First run will download Python environment (~15MB)

Spectrogram (Short-Time Fourier Transform)

We construct a signal consisting of a linear chirp (frequency sweeping from 10 Hz to 200 Hz) with an added high-frequency burst at $t = 1$ s. The STFT is computed manually using overlapping Hann-windowed segments, and the resulting spectrogram reveals the time-frequency structure: the chirp appears as a rising diagonal line, and the burst appears as a localized bright spot at 350 Hz.

import numpy as np
import matplotlib.pyplot as plt

# Chirp signal: frequency increases linearly
fs = 1000
duration = 2.0
t = np.arange(int(fs * duration)) / fs
f0, f1 = 10, 200
signal = np.sin(2*np.pi * (f0*t + (f1-f0)/(2*duration)*t**2))

# Add a burst
t_burst = 1.0
burst_idx = int(t_burst * fs)
signal[burst_idx:burst_idx+100] += 2*np.sin(2*np.pi*350*t[burst_idx:burst_idx+100])

# Simple STFT
window_size = 256
hop = 64
n_windows = (len(signal) - window_size) // hop
spectrogram = np.zeros((window_size // 2, n_windows))
window = np.hanning(window_size)

for i in range(n_windows):
    start = i * hop
    segment = signal[start:start+window_size] * window
    fft_result = np.abs(np.fft.fft(segment))[:window_size//2]
    spectrogram[:, i] = fft_result

freq_axis = np.linspace(0, fs/2, window_size//2)
time_axis = np.arange(n_windows) * hop / fs

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 8))
fig.suptitle('Short-Time Fourier Transform (Spectrogram)', color='white', fontsize=14)

ax1.plot(t, signal, color='#818cf8', linewidth=0.5)
ax1.set_title('Signal: Linear Chirp + Burst', color='#a5b4fc')
ax1.set_xlabel('Time (s)', color='#94a3b8')

im = ax2.pcolormesh(time_axis, freq_axis, 20*np.log10(spectrogram + 1e-10),
                      cmap='magma', shading='auto')
ax2.set_title('Spectrogram (dB)', color='#a5b4fc')
ax2.set_xlabel('Time (s)', color='#94a3b8')
ax2.set_ylabel('Frequency (Hz)', color='#94a3b8')
ax2.set_ylim(0, 400)
plt.colorbar(im, ax=ax2, label='Power (dB)')

for ax in [ax1, ax2]:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')

plt.tight_layout()
plt.show()

Click Run to execute the Python code

First run will download Python environment (~15MB)

Chapter Summary

From series to transform: Letting$T_0 \to \infty$ in the Fourier series converts the discrete harmonic sum into the continuous Fourier integral.
Forward/inverse pair: $X(f) = \int x(t)\,e^{-j2\pi ft}\,dt$ and$x(t) = \int X(f)\,e^{j2\pi ft}\,df$.
Properties: Linearity, time/frequency shift, scaling, convolution-multiplication duality, differentiation, integration, and duality.
Convolution theorem: Convolution in time equals multiplication in frequency, and vice versa.
Parseval/Plancherel: Energy is conserved across the transform; $\|x\|^2 = \|X\|^2$.
Uncertainty principle: $\sigma_t \cdot \sigma_\omega \geq 1/2$, with equality for Gaussians.
Transform pairs: rect-sinc, Gaussian-Gaussian, exponential-Lorentzian, delta-constant, cosine-delta pair.
STFT: Windowed Fourier analysis for time-varying signals; the spectrogram visualizes time-frequency content.

Signal Theory Course

Chapter 3: The Fourier Transform

3.1 From Series to Transform

3.2 The Continuous Fourier Transform

Existence Conditions

3.3 Key Properties

3.4 The Convolution Theorem

Proof Sketch

Multiplication-Convolution Duality

3.5 Plancherel / Parseval Theorem (Continuous)

3.6 Heisenberg Uncertainty Principle

Proof Outline

Connection to Quantum Mechanics

3.7 Common Transform Pairs

3.8 Short-Time Fourier Transform (STFT)

Time-Frequency Resolution Trade-off

Common Window Functions

Interactive Python Examples

Fourier Transform of Common Signals

Uncertainty Principle Demonstration

Spectrogram (Short-Time Fourier Transform)

Chapter Summary