Hardware Implementation of FFT Algorithm Based on FPGA

#fftalgorithm #fft #fpga #dsp

The Fast Fourier Transform (FFT) is a computationally efficient algorithm for computing the Discrete Fourier Transform (DFT), widely used in digital signal processing (DSP), communications, and real-time spectral analysis. Implementing FFT on FPGA (Field-Programmable Gate Array) offers high-speed, parallel processing advantages over software-based implementations.

1. Why FPGA for FFT?

Parallel Processing → FFT involves butterfly operations that can be executed simultaneously.
Low Latency → Hardware acceleration reduces computation time.
Real-Time Processing → Suitable for radar, 5G, and SDR (Software-Defined Radio).
Customizable Precision → Fixed-point or floating-point arithmetic can be optimized.

2. FFT Algorithm Overview
Radix-2 Decimation-in-Time (DIT) FFT

N-point FFT requires log₂N stages, each with N/2 butterfly operations.
Butterfly Structure:

text
X[k] = X_even[k] + W_N^k · X_odd[k]
X[k + N/2] = X_even[k] - W_N^k · X_odd[k]

where W_N^k = e^(-j2πk/N) (twiddle factor).

Radix-4 & Split-Radix FFT

Higher Radix reduces stages but increases complexity.
Split-Radix combines Radix-2 and Radix-4 for efficiency.

3. FPGA Implementation Steps
(1) Architecture Selection
Pipelined FFT (Systolic Array)

Each stage processes data in a pipeline (low latency, high throughput).
Example: Xilinx FFT IP Core.

Memory-Based FFT

Uses block RAM (BRAM) for storing intermediate results.
Slower but resource-efficient.

Iterative FFT

Reuses a single butterfly unit (low area, high latency).

(2) Fixed-Point vs. Floating-Point

(3) Key Hardware Components

Butterfly Processing Unit (BFU)

Performs complex multiply-accumulate (MAC) operations.
Optimized using DSP slices in FPGA.

Twiddle Factor ROM

Stores precomputed W_N^k values (sine/cosine).

Data Reordering (Bit-Reversal)

FFT output is in bit-reversed order → needs reordering.

Control FSM (Finite State Machine)

Manages data flow between stages.

(4) Optimization Techniques

Time-Multiplexing → Reuse hardware for multiple stages.
Burst-Mode Processing → Process data in blocks.
Parallel Butterfly Units → Increases speed at the cost of area.

4. FPGA Implementation Example (Verilog/VHDL)
Radix-2 Butterfly Unit (Verilog Snippet)

verilog
module butterfly (
  input signed [15:0] ar, ai, // Input (real/imag)
  input signed [15:0] br, bi, // Input (real/imag)
  input signed [15:0] wr, wi, // Twiddle factor (W_N^k)
  output signed [15:0] xr, xi, // Output X[k]
  output signed [15:0] yr, yi  // Output X[k + N/2]
);
  // Intermediate products
  wire signed [31:0] mult_r = (br * wr) - (bi * wi);
  wire signed [31:0] mult_i = (br * wi) + (bi * wr);

  // Truncate to 16-bit
  assign xr = ar + mult_r[30:15]; // X[k] = A + W·B
  assign xi = ai + mult_i[30:15];
  assign yr = ar - mult_r[30:15]; // X[k + N/2] = A - W·B
  assign yi = ai - mult_i[30:15];
endmodule

Twiddle Factor ROM (VHDL Snippet)

vhdl
library IEEE;
use IEEE.STD_LOGIC_1164.ALL;
use IEEE.NUMERIC_STD.ALL;

entity twiddle_rom is
  Port (
    addr : in std_logic_vector(3 downto 0);
    wr, wi : out signed(15 downto 0)
  );
end entity;

architecture Behavioral of twiddle_rom is
  type rom_type is array (0 to 15) of signed(15 downto 0);
  constant cos_rom : rom_type := (
    x"7FFF", x"7D8A", x"7641", x"6A6D", ... -- Precomputed cosine
  );
  constant sin_rom : rom_type := (
    x"0000", x"18F8", x"30FB", x"471C", ... -- Precomputed sine
  );
begin
  wr <= cos_rom(to_integer(unsigned(addr)));
  wi <= -sin_rom(to_integer(unsigned(addr))); // Negative for W_N^k
end architecture;

5. FPGA Resource Utilization