Matrix multiplication is the computational backbone of AI accelerators, and as neural networks grow in complexity, so does the need for larger and faster hardware modules. In this article, we’ll show you how to scale a pipelined Matrix Multiply Unit (MMU) from 2×2 to 4×4 matrices, discuss the design challenges, and provide a practical Verilog implementation with a testbench for verification.
Why Scale to 4×4 Matrices?
- Realistic Workloads: Most AI models use matrices much larger than 2×2. A 4×4 MMU is a practical step toward real-world applications.
- Performance: Larger MMUs can process more data per cycle, increasing throughput.
- Design Reusability: The techniques used here can be extended to even larger matrices or parameterized for flexibility.
Design Principles for a 4×4 Pipelined MMU
Scaling up means more multiplications and additions. For a 4×4 matrix multiplication:
- Inputs: Two 4×4 matrices (A and B)
- Outputs: One 4×4 result matrix (C)
- Computation: For each element C[i][j],
Pipelining Approach:
- Stage 1: Compute all 4 partial products for each of 16 output element (total 64 multiplications).
- Stage 2: Sum the partial products for each output element (16 additions).
Verilog Implementation: 4×4 Pipelined MMU
Here’s a scalable, pipelined Verilog module for 4×4 matrix multiplication:
module mmu_4x4_pipelined (
input clk,
input rst,
input [7:0] a [0:3][0:3], // Matrix A
input [7:0] b [0:3][0:3], // Matrix B
output [17:0] c [0:3][0:3] // Matrix C (result)
);
// Stage 1: Multiplication
reg [15:0] products [0:3][0:3][0:3]; // products[i][j][k] = A[i][k] * B[k][j]
integer i, j, k;
always @(posedge clk or posedge rst) begin
if (rst) begin
for (i=0; i<4; i=i+1)
for (j=0; j<4; j=j+1)
for (k=0; k<4; k=k+1)
products[i][j][k] <= 0;
end else begin
for (i=0; i<4; i=i+1)
for (j=0; j<4; j=j+1)
for (k=0; k<4; k=k+1)
products[i][j][k] <= a[i][k] * b[k][j];
end
end
// Stage 2: Addition
reg [17:0] c_reg [0:3][0:3];
always @(posedge clk or posedge rst) begin
if (rst) begin
for (i=0; i<4; i=i+1)
for (j=0; j<4; j=j+1)
c_reg[i][j] <= 0;
end else begin
for (i=0; i<4; i=i+1)
for (j=0; j<4; j=j+1)
c_reg[i][j] <= products[i][j][0] + products[i][j][1] +
products[i][j][2] + products[i][j][3];
end
end
// Output assignment
generate
genvar gi, gj;
for (gi=0; gi<4; gi=gi+1) begin: out_i
for (gj=0; gj<4; gj=gj+1) begin: out_j
assign c[gi][gj] = c_reg[gi][gj];
end
end
endgenerate
endmodule
Note:
- This code uses SystemVerilog-style arrays for clarity. If you’re using plain Verilog, you’ll need to flatten the arrays or use individual signals. However, we would suggest to use SystemVerilog-style arrays for ease of coding.
- The design is fully pipelined: new inputs can be loaded every clock cycle. First output is available after two clock cycles and following outputs are available every clock cycles.
Testbench for 4×4 Pipelined MMU
Here’s a simple testbench to verify the MMU:
module tb_mmu_4x4_pipelined;
reg clk, rst;
reg [7:0] a [0:3][0:3];
reg [7:0] b [0:3][0:3];
wire [17:0] c [0:3][0:3];
mmu_4x4_pipelined uut (
.clk(clk), .rst(rst), .a(a), .b(b), .c(c)
);
// Clock generation
initial clk = 0;
always #5 clk = ~clk;
integer i, j;
initial begin
rst = 1;
for (i=0; i<4; i=i+1) begin
for (j=0; j<4; j=j+1) begin
a[i][j] = 0;
b[i][j] = 0;
end
end
#12;
rst = 0;
// Test Case: Identity matrix multiplication
for (i=0; i<4; i=i+1) begin
for (j=0; j<4; j=j+1) begin
a[i][j] = (i == j) ? 1 : 0;
b[i][j] = (i == j) ? 1 : 0;
end
end
#10; // 1 clock cycle delay
// Test Case: All ones
for (i=0; i<4; i=i+1) begin
for (j=0; j<4; j=j+1) begin
a[i][j] = 1;
b[i][j] = 1;
end
end
#10; // 1 clock cycle delay
// First output after the intial delay of 2 clock cycles
$display("Identity Matrix Output:");
for (i=0; i<4; i=i+1) begin
$write("[ ");
for (j=0; j<4; j=j+1)
$write("%d ", c[i][j]);
$write("]\n");
end
#10; // 1 clock cycle delay
// Consecutive output after 1 clock cycle
$display("All Ones Output:");
for (i=0; i<4; i=i+1) begin
$write("[ ");
for (j=0; j<4; j=j+1)
$write("%d ", c[i][j]);
$write("]\n");
end
$finish;
end
endmodule
Design Tips for Larger Matrices
- Parameterization: Use Verilog parameters or SystemVerilog
parameter/localparamto make the matrix size configurable. - Resource Sharing: For very large matrices, consider sharing multipliers and adders to save area.
- Deeper Pipelining: For even higher throughput, break the addition stage into multiple pipeline stages.
Scaling a pipelined MMU to 4×4 matrices is a practical step toward real-world AI hardware. With pipelining, you achieve high throughput and efficient resource utilization. The techniques shown here can be extended to larger matrices, parameterized designs, and integrated into full AI accelerators.
Ready to go further? Try parameterizing the MMU for NxN matrices.
Have questions or want to share your own scalable MMU design? Leave a comment below!
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