DSP 101
M5 · L1
Module 5 — The Discrete Fourier Transform
From Time Domain
to Frequency Domain

Why every signal is a sum of sinusoids — and how the DTFT and DFT reveal the frequency content of discrete-time signals.

1 / 9
DSP 101
M5 · L1
Motivation
Every Signal is a
Sum of Sinusoids

Fourier's insight: any signal can be expressed as a weighted sum of sinusoids. The frequency domain reveals this decomposition, making filter design and noise analysis natural and intuitive.

Key advantage
Convolution in time = multiplication in frequency. Filtering becomes pointwise multiplication.
2 / 9
DSP 101
M5 · L1
The DTFT
Discrete-Time
Fourier Transform

The DTFT evaluates the Z-transform on the unit circle z = ejω. Result: a continuous, 2π-periodic function of normalized frequency ω (rad/sample).

DTFT
X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]\,e^{-j\omega n}
3 / 9
DSP 101
M5 · L1
Frequency Scale
Normalized
Frequency

Frequency ω is in radians per sample. To convert to Hz, multiply by fs/(2π).

ω = 0
DC (0 Hz)
ω = π
Nyquist fs/2
Conversion
ω = 2πf / fs — so f = 10 kHz at fs = 48 kHz gives ω ≈ 1.31 rad/sample
4 / 9
DSP 101
M5 · L1
DTFT Properties
Four Essential Properties
  • Linearity — DTFT{ax + by} = aX + bY
  • Time shift — delay n₀ samples → multiply by e−jωn₀
  • Convolution ↔ multiplication — x∗h ↔ X·H
  • Conjugate symmetry — real x[n] → |X(ejω)| is even
Most important
Convolution–multiplication duality: LTI filtering becomes pointwise spectral multiplication.
5 / 9
DSP 101
M5 · L1
The Problem
DTFT is Continuous
— Not Computable

X(ejω) takes infinitely many values. A digital machine needs a finite, discrete frequency representation. Solution: sample the DTFT at N equally spaced frequencies.

∞
DTFT values
N
DFT values
6 / 9
DSP 101
M5 · L1
The DFT
Sampling the DTFT:
The DFT

Evaluate the DTFT at ωk = 2πk/N for k = 0 … N−1. Result: N complex numbers, fully computable and invertible.

DFT
X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-j2\pi kn/N}
7 / 9
DSP 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

8 / 9
DSP 101
M5 · L1
Key Takeaways
What You Learned
  • Every signal = sum of sinusoids; frequency domain reveals the decomposition
  • DTFT maps x[n] → continuous X(ejω), periodic with period 2π
  • ω = π corresponds to Nyquist frequency fs/2
  • Convolution in time = multiplication in frequency
  • DTFT is not computable — sampling it at N points gives the DFT
  • DFT is periodic in time (circular): treats x[n] as one period of N
Read full article →
Next lesson All lessons
9 / 9
1 / 8 muted