How to perform fixed-point arithmetic on an FPGA?

三月 19, 2025

Performing fixed-point arithmetic on an FPGA is a common technique for achieving efficient and high-performance computations, especially in applications like digital signal processing (DSP), image processing, and control systems. Fixed-point arithmetic avoids the complexity and resource usage of floating-point arithmetic while providing sufficient precision for many applications. Here's a step-by-step guide to implementing fixed-point arithmetic on an FPGA:

1. Understand Fixed-Point Representation

Fixed-point numbers represent real numbers using a fixed number of integer and fractional bits. The format is typically denoted as Qm.n, where:

m: Number of integer bits (including the sign bit for signed numbers).
n: Number of fractional bits.

For example, a Q8.8 format uses 8 integer bits and 8 fractional bits, with a total of 16 bits.

2. Choose the Fixed-Point Format

Determine the range and precision required for your application.
Choose the number of integer bits (m) to cover the maximum value.
Choose the number of fractional bits (n) to achieve the desired precision.

Example:

For a range of [-128, 127] and a precision of 0.0039 (1/256), use Q8.8.

3. Implement Fixed-Point Arithmetic Operations

Fixed-point arithmetic involves scaling numbers and handling overflow/underflow. Below are the key operations:

Addition and Subtraction

Ensure both operands have the same Qm.n format.
Perform the operation directly, but handle overflow/underflow.

Example (Q8.8):

signal a, b, sum : signed(15 downto 0); -- Q8.8 format
sum <= a + b; -- Direct addition

Multiplication

Multiply two Qm.n numbers to get a Q(2m).(2n) result.
Truncate or round the result to the desired Qm.n format.

Example (Q8.8):

signal a, b : signed(15 downto 0); -- Q8.8 format
signal product : signed(31 downto 0); -- Q16.16 format
signal result : signed(15 downto 0); -- Q8.8 format

product <= a * b; -- Q16.16 result
result <= product(23 downto 8); -- Truncate to Q8.8

Division

Division is more complex and often avoided in FPGAs.
Use multiplication by the reciprocal or iterative algorithms (e.g., Newton-Raphson).

Scaling

To convert between different Qm.n formats, shift the bits left or right.
Example: Convert Q8.8 to Q12.4 by shifting left by 4 bits.

4. Handle Overflow and Underflow

Monitor the most significant bits (MSBs) to detect overflow/underflow.
Use saturation logic to clamp values within the representable range.

Example (Q8.8):

if sum > 32767 then
    sum <= 32767; -- Saturate at maximum value
elsif sum < -32768 then
    sum <= -32768; -- Saturate at minimum value
end if;

5. Use FPGA Resources Efficiently

DSP Slices: Use dedicated DSP slices for multiplication and addition.
LUTs: Implement smaller operations (e.g., addition, shifting) using LUTs.
Pipelining: Add pipeline stages to improve throughput and timing.

6. Implement in HDL

Use Hardware Description Languages (HDLs) like VHDL or Verilog to implement fixed-point arithmetic.

VHDL Example (Q8.8 Addition)

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

entity fixed_point_adder is
    Port (
        a : in signed(15 downto 0); -- Q8.8
        b : in signed(15 downto 0); -- Q8.8
        sum : out signed(15 downto 0) -- Q8.8
    );
end fixed_point_adder;

architecture Behavioral of fixed_point_adder is
begin
    sum <= a + b; -- Direct addition
end Behavioral;

Verilog Example (Q8.8 Multiplication)

module fixed_point_multiplier (
    input signed [15:0] a, // Q8.8
    input signed [15:0] b, // Q8.8
    output signed [15:0] result // Q8.8
);
    reg signed [31:0] product; // Q16.16

    always @(*) begin
        product = a * b; // Q16.16 result
        result = product[23:8]; // Truncate to Q8.8
    end
endmodule

7. Test and Verify

Simulate your design using testbenches to verify correctness.
Check for edge cases (e.g., overflow, underflow, rounding errors).

VHDL Testbench Example

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

entity tb_fixed_point_adder is
end tb_fixed_point_adder;

architecture Behavioral of tb_fixed_point_adder is
    signal a, b, sum : signed(15 downto 0);
begin
    uut: entity work.fixed_point_adder
        port map (a => a, b => b, sum => sum);

    process
    begin
        a <= x"0100"; -- 1.0 in Q8.8
        b <= x"0200"; -- 2.0 in Q8.8
        wait for 10 ns;
        assert sum = x"0300" report "Test failed" severity error;

        wait;
    end process;
end Behavioral;

8. Optimize for Performance

Use pipelining to improve clock speed.
Minimize resource usage by sharing hardware for multiple operations.
Use vendor-specific IP cores for complex operations (e.g., Xilinx DSP48).

9. Use High-Level Synthesis (HLS) Tools

For faster development, use HLS tools like Xilinx Vivado HLS or Intel HLS to write fixed-point algorithms in C/C++ and automatically generate HDL code.

By following these steps, you can efficiently implement fixed-point arithmetic on an FPGA, balancing precision, performance, and resource usage for your specific application.

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