Continuous-Time Fourier Transform

Introduction

The Continuous-Time Fourier Transform (CTFT) extends the Fourier series from periodic signals to aperiodic (non-periodic) signals. While the Fourier series decomposes periodic signals into discrete harmonic frequencies, the CTFT decomposes any signal into a continuous spectrum of frequencies. This is one of the most important mathematical tools in engineering — it tells us the frequency content of any signal, whether periodic or not.

For B.Tech students, the CTFT is the bridge between time-domain and frequency-domain descriptions of signals. Once you understand this transform, you can analyze how signals pass through systems, understand bandwidth, design filters, and work with communication systems.

From Fourier Series to Fourier Transform

Consider a periodic signal with period $T_0$ and Fourier coefficients $c_n$. As $T_0 \to \infty$ (signal becomes aperiodic):

• The harmonic spacing $\omega_0 = 2\pi/T_0$ shrinks to zero
• Discrete frequencies become a continuous variable $\omega$
• The sum becomes an integral
• $T_0 c_n$ converges to a continuous function $X(\omega)$

This limiting process yields the Fourier transform pair.

Definition: The Transform Pair

Forward Transform (Analysis): $$X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$$

Inverse Transform (Synthesis): $$x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega) e^{j\omega t} d\omega$$

The forward transform decomposes $x(t)$ into complex exponential components. $X(\omega)$ is the spectral density — it tells us how much of each frequency $\omega$ is present in $x(t)$.

In terms of cyclic frequency $f$ (Hz): $$X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$$ $$x(t) = \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df$$

Existence Conditions

The Fourier transform exists if at least one of these sufficient conditions holds:

1. Absolute integrability: $\int_{-\infty}^{\infty}|x(t)|dt < \infty$ (sufficient for pointwise convergence)
2. Finite energy: $\int_{-\infty}^{\infty}|x(t)|^2 dt < \infty$ (sufficient for mean-square convergence)
3. Generalized functions: Signals like $u(t)$, $\cos(\omega_0 t)$, and $\delta(t)$ have transforms in the distributional sense using impulse functions

Most engineering signals satisfy at least one of these conditions.

Interpreting the Fourier Transform

$X(\omega)$ is generally complex-valued: $$X(\omega) = |X(\omega)|e^{j\angle X(\omega)}$$

• Magnitude spectrum $|X(\omega)|$: How much of frequency $\omega$ is present
• Phase spectrum $\angle X(\omega)$: The phase (timing) of each frequency component
• Power spectral density (for power signals): $S(\omega) = |X(\omega)|^2$

Important Transform Pairs

Rectangular Pulse

$$\text{rect}(t/T) \xleftrightarrow{FT} T\text{sinc}(fT)$$

Exponential Decay

$$e^{-at}u(t) \xleftrightarrow{FT} \frac{1}{a + j\omega}, \quad a > 0$$

Magnitude: $|X(\omega)| = 1/\sqrt{a^2 + \omega^2}$ (Lorentzian shape, low-pass)

Double-Sided Exponential

$$e^{-a|t|} \xleftrightarrow{FT} \frac{2a}{a^2 + \omega^2}$$

Real and even (because the time signal is real and even).

Gaussian

$$e^{-\pi t^2} \xleftrightarrow{FT} e^{-\pi f^2}$$

A Gaussian transforms to a Gaussian — the only function that is its own Fourier transform!

Impulse

$$\delta(t) \xleftrightarrow{FT} 1$$

The impulse contains ALL frequencies equally — a flat spectrum.

Constant (DC)

$$1 \xleftrightarrow{FT} 2\pi\delta(\omega)$$

A constant signal has all its energy at zero frequency.

Cosine

$$\cos(\omega_0 t) \xleftrightarrow{FT} \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$$

A pure cosine has energy only at $\pm\omega_0$.

Symmetry Properties

For a real signal $x(t)$:

• $X(-\omega) = X^*(\omega)$ (conjugate symmetry)
• $|X(-\omega)| = |X(\omega)|$ (even magnitude)
• $\angle X(-\omega) = -\angle X(\omega)$ (odd phase)
• If $x(t)$ is also even: $X(\omega)$ is real and even
• If $x(t)$ is odd: $X(\omega)$ is purely imaginary and odd

Energy Spectral Density and Parseval's Theorem

Parseval's theorem (energy conservation): $$E = \int_{-\infty}^{\infty}|x(t)|^2 dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|X(\omega)|^2 d\omega$$

The total energy computed in the time domain equals the total energy in the frequency domain. The function $|X(\omega)|^2$ is the energy spectral density — it shows how energy is distributed across frequencies.

Bandwidth

The bandwidth of a signal quantifies its frequency extent:

• 3-dB bandwidth: frequency range where $|X(f)|$ is within $1/\sqrt{2}$ of its peak
• Null-to-null bandwidth: distance between first zeros of $|X(f)|$
• 99% energy bandwidth: range containing 99% of total energy

For a rectangular pulse of width $T$: null bandwidth = $1/T$. Shorter pulses → wider bandwidth (uncertainty principle).

Worked Examples

Example 1: Find the FT of $x(t) = e^{-2t}u(t)$.

$$X(\omega) = \int_0^{\infty} e^{-2t}e^{-j\omega t}dt = \int_0^{\infty}e^{-(2+j\omega)t}dt = \frac{1}{2+j\omega}$$

Magnitude: $|X(\omega)| = 1/\sqrt{4+\omega^2}$, Phase: $\angle X(\omega) = -\arctan(\omega/2)$

Example 2: Find the FT of $x(t) = e^{-|t|}$.

$$X(\omega) = \int_{-\infty}^{0}e^{t}e^{-j\omega t}dt + \int_0^{\infty}e^{-t}e^{-j\omega t}dt = \frac{1}{1+j\omega} + \frac{1}{1-j\omega} = \frac{2}{1+\omega^2}$$

Real and even (as expected since $x(t)$ is real and even).

Key Takeaways

• The CTFT extends Fourier series to aperiodic signals: discrete spectrum → continuous spectrum
• Transform pair: $X(\omega) = \int x(t)e^{-j\omega t}dt$ and $x(t) = \frac{1}{2\pi}\int X(\omega)e^{j\omega t}d\omega$
• $|X(\omega)|$ is the magnitude spectrum; $\angle X(\omega)$ is the phase spectrum
• Parseval's theorem: signal energy equals integrated spectral energy
• Key pairs: rect↔sinc, exponential↔Lorentzian, Gaussian↔Gaussian, δ↔1
• Real signals have conjugate-symmetric spectra; real-even signals have real-even spectra
• Time-bandwidth product is bounded below (uncertainty principle): narrow in time → wide in frequency