3.1 Definition and Derivation of Fourier Transform

The Fourier Transform converts a continuous-time signal into a representation of its frequency content. This is one of the most fundamental operations in signal processing because it lets you analyze, filter, and manipulate signals by working with their frequency components rather than their raw time-domain samples.

The Fourier Transform

Definition and Properties

The Fourier Transform decomposes a continuous-time signal into its constituent frequencies, producing a frequency-domain representation. The forward transform maps a time-domain signal $x(t)$ to its frequency-domain counterpart $X(\omega)$, where $\omega$ is angular frequency in radians per second:

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

Here, $j$ is the imaginary unit and $e^{-j\omega t}$ is a complex exponential that acts as a "probe" for frequency $\omega$. The integral essentially measures how much of that frequency is present in $x(t)$.

The inverse Fourier Transform recovers the original time-domain signal from $X(\omega)$:

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

Notice the $\frac{1}{2\pi}$ factor in front. This normalization appears because we're using angular frequency $\omega$ (radians/second). If you use ordinary frequency $f$ in Hz instead, the $2\pi$ factor disappears from the front but shows up inside the exponential as $e^{j2\pi ft}$.

Existence and Applicability

Not every signal has a Fourier Transform. The most common sufficient condition is absolute integrability (also called the Dirichlet condition):

$\int_{-\infty}^{\infty} |x(t)| \, dt < \infty$

This guarantees the integral defining $X(\omega)$ converges to a finite value for every $\omega$.

Signals with finite energy also have well-defined Fourier Transforms:

$\int_{-\infty}^{\infty} |x(t)|^2 \, dt < \infty$

A finite-energy signal might not be absolutely integrable (for example, a sinc function), but its transform still exists in the mean-square sense.

Signals like exponential decays, Gaussian pulses, and rectangular pulses all satisfy these conditions. Purely periodic signals like $\cos(\omega_0 t)$ technically don't satisfy either condition, but their transforms can still be defined using Dirac delta functions, which you'll encounter as you go deeper.

Deriving the Fourier Transform

Fourier Series and Its Limitations

The Fourier Series represents a periodic signal with period $T$ as a sum of complex exponentials at harmonic frequencies $n\omega_0$, where $\omega_0 = \frac{2\pi}{T}$ is the fundamental frequency. The coefficients are:

$c_n = \frac{1}{T} \int_{-T/2}^{T/2} x(t)e^{-jn\omega_0 t} \, dt$

Each $c_n$ tells you the amplitude and phase of the $n$-th harmonic. The full reconstruction is $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$.

The limitation is that this only works for periodic signals. If a signal doesn't repeat, there's no period $T$ to define, and the series doesn't apply directly.

Transition from Fourier Series to Fourier Transform

The Fourier Transform can be derived by taking the Fourier Series and letting the period grow to infinity. Here's how the argument works:

1. Start with a periodic signal of period $T$ and its Fourier Series coefficients $c_n$.
2. Define $X_T(\omega) = T \cdot c_n$ evaluated at $\omega = n\omega_0$. Substituting the formula for $c_n$, this gives $X_T(n\omega_0) = \int_{-T/2}^{T/2} x(t)e^{-jn\omega_0 t} \, dt$.
3. As $T \to \infty$, the fundamental frequency $\omega_0 = \frac{2\pi}{T}$ shrinks toward zero. The discrete harmonic frequencies $n\omega_0$ become increasingly dense along the frequency axis.
4. The spacing between adjacent harmonics, $\Delta\omega = \omega_0$, becomes the infinitesimal $d\omega$. The summation over $n$ becomes an integral over $\omega$.
5. The integration limits $-T/2$ to $T/2$ extend to $-\infty$ to $\infty$, and you arrive at $X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} \, dt$.

The key insight: the Fourier Series gives you a discrete set of frequency components (harmonics), while the Fourier Transform gives you a continuous spectrum. The transform is the natural generalization of the series to aperiodic signals.

Time vs Frequency Domains

Duality and Interplay

The Fourier Transform pairs every time-domain signal $x(t)$ with a frequency-domain representation $X(\omega)$. These are two views of the same signal:

• Time domain $x(t)$: shows how the signal's amplitude changes over time.
• Frequency domain $X(\omega)$: shows which frequencies are present and how strong each one is.

Operations in one domain have direct counterparts in the other. For example:

• Time shift: delaying $x(t)$ by $t_0$ multiplies $X(\omega)$ by $e^{-j\omega t_0}$ (changes phase, not magnitude).
• Time scaling: compressing a signal in time spreads it out in frequency, and vice versa.
• Convolution in time corresponds to multiplication in frequency, which is why filtering is so natural in the frequency domain.

Reversibility and Reconstruction

Because the forward and inverse transforms are both well-defined (given the existence conditions), you can freely move between domains without losing information. This is what makes frequency-domain processing practical:

1. Transform the signal to the frequency domain using the forward Fourier Transform.
2. Modify $X(\omega)$ as needed (e.g., zero out unwanted frequency ranges to filter noise).
3. Transform back to the time domain using the inverse Fourier Transform.

The reconstructed signal is identical to what you'd get by performing the equivalent (and often much harder) operation directly in the time domain.

Physical Interpretation of the Fourier Transform

Frequency Content and Spectral Analysis

The output $X(\omega)$ is generally complex-valued, so it carries two pieces of information at each frequency:

• Magnitude spectrum $|X(\omega)|$: the amplitude (strength) of each frequency component.
• Phase spectrum $\angle X(\omega)$: the phase offset of each frequency component.

Together, these fully characterize the signal's frequency content. Spectral analysis means examining $|X(\omega)|$ and $\angle X(\omega)$ to extract useful information. For instance, you can identify the dominant frequency of a vibrating structure, measure the bandwidth of a communication signal, or detect harmonics in an audio recording.

Applications in Signal Processing

The frequency-domain perspective enables several core signal processing tasks:

• Filtering: Design low-pass, high-pass, or band-pass filters by selectively keeping or attenuating frequency ranges in $X(\omega)$. A low-pass filter, for example, sets $X(\omega) = 0$ for $|\omega|$ above some cutoff.
• Denoising: Techniques like spectral subtraction estimate the noise spectrum and subtract it from the signal's spectrum before inverting back to the time domain.
• Compression: Algorithms like MP3 (audio) and JPEG (images) exploit the fact that much of a signal's energy is concentrated in a relatively small number of frequency components. By discarding weak components, you achieve significant data reduction with minimal perceptual loss.

The physical meaning of $\omega$ depends on context. For time-domain signals, $\omega$ represents temporal frequency. For spatial signals (like images), $\omega$ represents spatial frequency. The math is the same either way.