Fourier Transform: Detailed Notes
1 Introduction to Fourier Transform
The Fourier Transform is a mathematical tool used to decompose a function (often a
time-domain signal) into its constituent frequencies. It transforms a function of time,
f (t), into a function of frequency, F (ω ), revealing the frequency components that make
up the original signal. This is particularly useful in signal processing, communications,
image processing, and many other fields.
The Fourier Transform is named after Jean-Baptiste Joseph Fourier, who introduced the
idea that any periodic function can be represented as a sum of sine and cosine functions.
2 Definition
The Fourier Transform of a continuous-time signal f (t) is defined as:
∫ ∞
F (ω ) = f (t)e− jωt dt
−∞
where:
• F (ω ) is the Fourier Transform, a complex-valued function of angular frequency ω.
• f (t) is the time-domain signal.
• e− jωt = cos(ωt) − j sin(ωt) is the complex exponential.
• j is the imaginary unit (j2 = −1).
The inverse Fourier Transform reconstructs the original signal:
∫ ∞
1
f (t) = F (ω )e jωt dω
2π −∞
For discrete signals, the Discrete Fourier Transform (DFT) is used, and its fast imple-
mentation is known as the Fast Fourier Transform (FFT).
3 Properties of Fourier Transform
The Fourier Transform has several important properties that make it versatile:
1
1. Linearity: For constants a and b, and functions f (t) and g(t),
F { a f (t) + bg(t)} = aF (ω ) + bG (ω )
2. Time Shifting: A shift in the time domain corresponds to a phase shift in the
frequency domain:
F { f (t − t0 )} = e− jωt0 F (ω )
3. Frequency Shifting: A shift in the frequency domain corresponds to modulation in
the time domain:
F {e jω0 t f (t)} = F (ω − ω0 )
4. Scaling: Scaling the time axis affects the frequency axis:
1 (ω)
F { f ( at)} = F
| a| a
5. Convolution Theorem: Convolution in the time domain corresponds to multiplica-
tion in the frequency domain:
F { f (t) ∗ g(t)} = F (ω ) · G (ω )
6. Differentiation: Differentiation in the time domain corresponds to multiplication
by jω in the frequency domain:
{ }
d
F f (t) = jωF (ω )
dt
4 Common Fourier Transform Pairs
Some common functions and their Fourier Transforms include:
• Rectangular pulse: f (t) = rect(t/T ) =⇒ F (ω ) = Tsinc(ωT/2).
• Dirac delta function: f (t) = δ(t) =⇒ F (ω ) = 1.
• Exponential decay: f (t) = e− at u(t) =⇒ F (ω ) = 1
a+ jω , for a > 0, where u(t) is
the unit step function.
√
• Gaussian: f (t) = e−t =⇒ F (ω ) = πe−ω /4 .
2 2
5 Applications of Fourier Transform
The Fourier Transform is widely used in various fields:
• Signal Processing: To analyze the frequency content of signals, filter noise, or com-
press data.
• Communications: For modulation and demodulation of signals in systems like
AM/FM radio or digital communications.
2
• Image Processing: For tasks like image compression (JPEG), filtering, and edge
detection.
• Physics: To solve differential equations in quantum mechanics and wave propaga-
tion.
• Data Analysis: To analyze time-series data, such as stock prices or climate data.
6 Example
Consider a signal f (t) = e−|t| . Its Fourier Transform is:
∫ ∞
F (ω ) = e−|t| e− jωt dt
−∞
Split the integral at t = 0:
∫ 0 ∫ ∞
t − jωt
F (ω ) = ee dt + e−t e− jωt dt
−∞ 0
∫ 0 ∫ ∞
t(1− jω )
= e dt + e−t(1+ jω ) dt
−∞ 0
Evaluate each integral:
∫ 0
[ ]0
et(1− jω ) 1
et(1− jω ) dt = =
−∞ 1 − jω 1 − jω
−∞
∫ ∞
[ ]∞
−t(1+ jω ) e−t(1+ jω ) 1
e dt = − =
0 1 + jω 1 + jω
0
Thus:
1 1 2
F (ω ) = + =
1 − jω 1 + jω 1 + ω2
7 Conclusion
The Fourier Transform is a powerful tool that bridges the time and frequency domains,
enabling the analysis and manipulation of signals in countless applications. Understand-
ing its properties and applications is essential for fields like engineering, physics, and data
science.
