Dft and fft

1. Discrete Fourier Transform (DFT)
and FFT Algorithms
• Subtopics:
• - DTFT vs DFT
• - DFT Definition & Properties
• - DFT and Circular Convolution
• - FFT (Radix-2) Algorithms

2. DTFT and DFT – Relationship
• - DTFT: Infinite-length signal, continuous
frequency spectrum
• - DFT: Finite-length signal, discrete frequency
spectrum
• Relationship:
• DFT samples the DTFT at ω_k = 2πk/N

3. DFT Definition
• DFT:
• X[k] = Σ (n=0 to N-1) x[n] e^(-j2πkn/N)
• Inverse DFT:
• x[n] = (1/N) Σ (k=0 to N-1) X[k] e^(j2πkn/N)

4. DFT Properties
• 1. Periodicity:
• X[k+N] = X[k], x[n+N] = x[n]
• 2. Linearity:
• DFT{a·x[n] + b·y[n]} = a·X[k] + b·Y[k]
• 3. Symmetry:
• For real x[n], X[N-k] = X*[k]

5. DFT as a Linear Transformation
• DFT is a matrix-vector multiplication:
• X = W_N · x
• W_N is the DFT matrix transforming time-
domain to frequency-domain

6. Multiplication & Circular
Convolution
• - Time-domain circular convolution ↔
Frequency-domain multiplication
• - z[n] = Σ (m=0 to N-1) x[m] · y[(n - m) mod N]

7. Efficient Computation – Why FFT?
• - DFT direct computation: O(N²)
• - FFT algorithm: O(N log₂N)
• Advantage: Drastically reduces computation
time

8. Radix-2 FFT Algorithms
• Decimation-in-Time (DIT):
• - Split input into even/odd samples
• - Recursive
• Decimation-in-Frequency (DIF):
• - Split DFT output
• - Butterfly structure

9. Butterfly Diagram
• Butterfly structure:
• X[k] = x₁ + W_N^k·x₂
• X[k+N/2] = x₁ - W_N^k·x₂
• Used in both DIT and DIF FFT algorithms

10. Summary
• - DFT: Finite-length frequency analysis
• - Related to DTFT by frequency sampling
• - FFT (DIT & DIF): Fast DFT computation
• - Widely used in DSP and communications