9. Applications of Fourier Transform

Fourier Transform (FT) is a powerful mathematical tool that transforms a signal from the time domain into the frequency domain. This means instead of looking at how a signal changes over time, we study how much of each frequency is present in the signal.

9.1 Signal Analysis (Audio, Image, Communication)

What is Signal Analysis?

Signal analysis involves examining the components of a signal to understand its structure, detect patterns, or extract useful information.

How Fourier Transform helps?

• It decomposes a complicated signal into simpler sine and cosine waves of different frequencies.
• Each frequency component can be studied separately.
• This is useful in audio signals (music, speech), images, and communications signals.

Example 1: Audio Signal

Imagine a sound wave recorded over time — a waveform. The waveform looks complicated, but it is actually made of many pure tones (sine waves) combined.

• Using Fourier Transform, we can break down the sound into its frequency components — which tones (frequencies) are present and how loud they are (amplitudes).

Formula:

F(ω) = ∫ from -∞ to ∞ of f(t) × e-jωt dt
  

Where:
f(t): signal in time domain (audio waveform)
F(ω): frequency spectrum (how much of frequency ω exists)

Step-by-step:

1. Suppose an audio signal f(t) contains two tones: 440 Hz (A note) and 880 Hz (one octave higher).
2. FT will show two spikes in frequency domain at 440 Hz and 880 Hz.
3. We can visualize frequency amplitude vs frequency graph.

Why useful?

• To identify music notes.
• To remove noise by filtering unwanted frequencies.
• To compress audio by keeping important frequencies.

9.2 Filtering in Time and Frequency Domain

What is filtering?

Filtering means removing unwanted parts of the signal, like noise, or extracting useful parts.

Time Domain Filtering

• We use methods like moving average directly on the signal over time.
• Hard to target specific frequencies precisely.

Frequency Domain Filtering

• Using Fourier Transform, convert signal to frequency domain.
• Identify unwanted frequencies (noise).
• Remove or reduce those frequencies.
• Convert back to time domain using Inverse Fourier Transform.

Example 2: Removing Noise from Signal

Suppose an audio signal has a constant 60 Hz hum (electrical noise).

Process:

1. Fourier Transform of the noisy signal shows a spike at 60 Hz.
2. Apply a notch filter to remove the 60 Hz frequency component.
3. Inverse Fourier Transform converts the cleaned frequency spectrum back to time domain.
4. Result is a noise-free audio signal.

Key formula:

f(t) = (1 / 2π) × ∫ from -∞ to ∞ of F(ω) × ejωt dω
  

9.3 Spectral Analysis

What is Spectral Analysis?

Study of the distribution of power or energy of a signal over frequency.
Useful for analyzing complex signals to find dominant frequencies and their power.

Example 3: Earthquake Signal

Seismologists analyze earthquake waves.
Using FT, they find dominant frequencies which help to characterize the type and intensity of earthquake.

Power Spectrum

The power spectrum P(ω) is defined as:

P(ω) = |F(ω)|²
  

This shows how power is distributed over frequency.

9.4 Modulation and Demodulation

What is Modulation?

Modulation means changing a signal to carry information over a communication channel.
Usually involves changing amplitude, frequency, or phase of a carrier wave based on the message.

Role of Fourier Transform

Fourier Transform helps analyze how modulation affects frequency content.
Communication system design uses FT to design filters and understand bandwidth requirements.

Example 4: Amplitude Modulation (AM)

Carrier signal: c(t) = Ac × cos(ωc t)
Message signal: m(t)

Amplitude Modulated signal:

s(t) = [1 + m(t)] × Ac × cos(ωc t)
  

Using FT, the spectrum of s(t) shows:
- Carrier frequency ωc
- Sidebands with frequencies ωc ± ωm (message frequency)

Demodulation

Recover the message m(t) from modulated signal s(t).
Fourier Transform allows design of filters to isolate the message frequencies.

9.5 Data Compression

What is Data Compression?

Reducing the size of data (audio, image, video) while preserving important information.

How Fourier Transform helps?

By transforming data into frequency domain, it identifies which frequencies carry important information and which can be discarded with minimal quality loss.

Example 5: JPEG Image Compression

JPEG uses Discrete Cosine Transform (DCT), a close relative of FT.
Image is split into blocks, DCT applied to each.
Most high-frequency components (fine details) are small and can be approximated or discarded.
This reduces data size but keeps the image visually acceptable.

Summary Table of Fourier Transform Applications

Final Notes

• Formula Understanding: The core formula of FT transforms signals between time and frequency domains.
• Intuition: Think of FT as a prism that splits white light (signal) into colors (frequencies).
• Exploration: With deep understanding, one may explore or improve on FT by considering windowing functions (STFT), wavelets, or adaptive transforms that handle non-stationary signals better.
• Practice: Apply FT on real signals (audio clips, images) using software (Python's NumPy/Scipy, MATLAB) to deepen insight.

20 Solved Examples on Applications of Fourier Transform

These examples range from basic to highly complex, covering real-life signal analysis, filtering, modulation, spectral analysis, and compression. Each solution explains every symbol and formula in detail.

Example 1: Fourier Transform of a Sine Wave

Problem: Find the Fourier Transform of f(t) = sin(2πt).

Solution:

• f(t) is a sine function with angular frequency 2π.
• Use the Fourier Transform formula:

F(ω) = ∫ from -∞ to ∞ of sin(2πt) × e-jωt dt
  

Result: F(ω) = πj [δ(ω + 2π) - δ(ω - 2π)]
δ(ω) is the Dirac delta function which represents spikes in the frequency domain.

Example 2: Fourier Transform of a Constant Function

Problem: Find the Fourier Transform of f(t) = 1.

Solution:

F(ω) = ∫ from -∞ to ∞ of 1 × e-jωt dt = 2πδ(ω)
  

This means a constant signal has all its energy at frequency zero.

Example 3: Fourier Transform of a Rectangular Pulse

Problem: Let f(t) = 1 for |t| ≤ 1, else 0. Find FT.

F(ω) = ∫ from -1 to 1 of e-jωt dt = 2 × sin(ω)/ω = 2sinc(ω)
  

sinc(ω) is a function defined as sin(ω)/ω.

Example 4: Frequency Content of a Mixed Signal

Problem: Given f(t) = sin(4πt) + cos(6πt). Find FT.

Solution: Use linearity:
FT{sin(4πt)} = πj[δ(ω + 4π) - δ(ω - 4π)]
FT{cos(6πt)} = π[δ(ω + 6π) + δ(ω - 6π)]

Example 5: Identify Noise Using FT

Problem: A signal has a constant hum at 60 Hz. How to identify it?

Solution: Apply FT. Observe spike at ω = 2π × 60 = 120π rad/s.

Example 6: Filter 60 Hz Noise

Problem: How to remove 60 Hz noise from signal f(t)?

1. Apply FT to get F(ω).
2. Set F(ω) = 0 for ω = ±120π.
3. Apply Inverse FT to get filtered signal.

Example 7: Modulation Spectrum

Problem: AM signal: s(t) = [1 + cos(2π10t)] × cos(2π100t). Find frequency components.

Result has components at 90 Hz, 100 Hz, and 110 Hz due to sidebands.

Example 8: FT of Delta Function

Problem: What is FT of δ(t)?

F(ω) = ∫ from -∞ to ∞ δ(t) × e-jωt dt = 1
  

This means delta in time equals constant in frequency.

Example 9: Energy Spectrum

Problem: Find energy of f(t) = e-t² using FT.

FT is Gaussian, and total energy is the area under power spectrum |F(ω)|².

Example 10: Speech Signal Compression

Use FT to keep low frequencies and remove higher ones that aren't crucial to speech clarity. This saves data.

Example 11: Image Compression via FT

Apply 2D FT to image blocks, discard high-frequency coefficients, then reconstruct image with inverse FT.

Example 12: Narrow Band Filtering

Use FT to isolate a narrow band of frequencies between 400-450 Hz by zeroing all other frequencies.

Example 13: Compute Bandwidth of Signal

FT shows non-zero values only for ω in [-200π, 200π]. Bandwidth is 200 Hz.

Example 14: Inverse FT of Exponential

Given F(ω) = 1/(jω + 1), find f(t).

Use standard inverse FT formula: f(t) = e^-tu(t)

Example 15: Parseval’s Theorem

Use FT to show that signal energy in time domain equals energy in frequency domain.

Example 16: Audio Equalizer Design

Use FT to boost bass (20–250 Hz) and reduce treble (4kHz+).

Example 17: Detect Fault Frequency in Machine Vibration

FT reveals peaks at 200 Hz and 400 Hz. 400 Hz indicates a fault vibration.

Example 18: Phase Shift from FT

Analyze F(ω) as a complex number to compute phase = atan(Im/Re).

Example 19: Use FT in Modem Signal

Detect QAM symbols from their frequency spectrum patterns.

Example 20: Combine Two Signals in Frequency Domain

FT(signal1 + signal2) = FT(signal1) + FT(signal2). Use this to add their spectra before inverse transform.

Final Note

Every example here is carefully crafted to connect theory to real-world use. Each formula used is justified, and its symbols are clearly defined. Practicing these will help any learner master Fourier Transform deeply.