Bit Shifter

This module is a building block for application-specific shifts and rotates. It synthesizes to LUT logic and can be quite large if not specialized to a particular situation.

A Warning About Shift Right and Signed Division

While a left shift by N is always equivalent to a multiplication by 2^N for both signed and unsigned binary integers, an arithmetic shift right by N is only a truncating division by 2^N for positive binary integers. For negative integers, the result is so-called modulus division, and the quotient ends up off by one in magnitude, and must be corrected by adding +1, but only if an odd number results as part of the intermediate division steps. That is, if a non-zero bit was shifted out to the right.

To implement proper signed division by a power-of-two, use the Integer Divider by Power of Two module, which performs the correction for negative numbers and also calculates the remainder as a bonus.

Usage

We can treat the shift_amount and the shift_direction together as a signed magnitude number: the amount is an absolute value, and the direction is the sign of the value. Here, a shift_direction of 1, meaning a negative number, shifts to the right. Choosing this convention for the sign matches the behaviour of a shift when we think about it as a multiplication or division by a power of 2:

Multipliying by 8 is equivalent to 2³N, which is a shift-left by 3 steps.
Dividing N by 4 is equivalent to 2^-2N, which is a shift-right by 2 steps.

Adding together these multiples and fractions generated by the shifts enables the creation of small, cheap scaling by constant ratios:

3N = N + 2¹N
10N = 8N + 2N = 2³N + 2¹N
5N/4 = N + N/4 = N + 2^-2N
etc...

When the shift values are constant, the shifter reduces to simple rewiring, which in turn reduces the above examples to an adder or two each.

The shifts are internally unsigned and word_in and word_out are extended to the left and right so new bits can be shifted in and current bits shifted out without loss, regardless of shift amount or direction, which enables the creation of more complex shifts or rotates:

Wire the most-significant bit (MSB) of word_in to all word_in_left inputs and zero to all word_in_right inputs to create a signed arithmetic shift.
Wire the word_in MSB to word_in_right MSB (or vice-versa) to create a rotate function.
Feed word_out_left and word_out to a double-word adder and set the shift to +1 (left by 1) as part of the construction of a conditional-add multiplier, which multiplies two N-bit words in N cycles, giving a 2N-bit result.

`default_nettype none

module Bit_Shifter
#(
    parameter   WORD_WIDTH  = 0
)
(
    input   wire    [WORD_WIDTH-1:0]    word_in_left,
    input   wire    [WORD_WIDTH-1:0]    word_in,
    input   wire    [WORD_WIDTH-1:0]    word_in_right,

    input   wire    [WORD_WIDTH-1:0]    shift_amount,
    input   wire                        shift_direction, // 0/1 -> left/right

    output  reg     [WORD_WIDTH-1:0]    word_out_left,
    output  reg     [WORD_WIDTH-1:0]    word_out,
    output  reg     [WORD_WIDTH-1:0]    word_out_right
);

Let's document the shift direction convention again here, and define our initial values for the outputs and the intermediate result.

    localparam  LEFT_SHIFT  = 1'b0;
    localparam  RIGHT_SHIFT = 1'b1;

    localparam  TOTAL_WIDTH = WORD_WIDTH * 3;
    localparam  TOTAL_ZERO  = {TOTAL_WIDTH{1'b0}};
    localparam  WORD_ZERO   = {WORD_WIDTH{1'b0}};

    initial begin
        word_out_left    = WORD_ZERO;
        word_out         = WORD_ZERO;
        word_out_right   = WORD_ZERO;
    end 

    reg [TOTAL_WIDTH-1:0] word_in_total = TOTAL_ZERO;

Rather than do arithmetic and calculate slices of vectors to figure out where the shifted bits end up, let's concatenate the input words into one triple-wide word, shift it as an unsigned number, then deconcatenate the result into each output word. All we have to do is keep the same convention on bit significance: here LSB is on the right.

    always @(*) begin
        word_in_total = {word_in_left, word_in, word_in_right};
        {word_out_left, word_out, word_out_right} = (shift_direction == LEFT_SHIFT) ? word_in_total << shift_amount : word_in_total >> shift_amount; 
    end

endmodule

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