## Step 1: Entity Declaration

To begin, let's declare the entity for our 4-bit by 4-bit multiplier. The entity will define the input and output ports of the multiplier.

```vhdl

library IEEE;

use IEEE.STD_LOGIC_1164.ALL;

use IEEE.STD_LOGIC_ARITH.ALL;

use IEEE.STD_LOGIC_UNSIGNED.ALL;

entity multiplier_4x4 is

Port (

A : in std_logic_vector(3 downto 0);

B : in std_logic_vector(3 downto 0);

Result : out std_logic_vector(7 downto 0)

);

end entity;

```

**Explanation:
**

- We declare the entity `multiplier_4x4`.
- Our multiplier has three ports:
- `A`: A 4-bit input representing the first multiplicand.
- `B`: A 4-bit input representing the second multiplicand.
- `Result`: An 8-bit output representing the product of A and B.

## Step 2: Architecture Implementation

Now, let's dive into the behavior of our multiplier by defining the architecture using a process.

```vhdl

architecture Behavioral of multiplier_4x4 is

begin

process(A, B)

variable temp : std_logic_vector(7 downto 0) := (others => '0');

begin

for i in 0 to 3 loop

for j in 0 to 3 loop

if A(i) = '1' and B(j) = '1' then

temp(i + j) := '1'; -- Add the corresponding products to temp

end if;

end loop;

end loop;

Result <= temp; -- Assign the result to the output port

end process;

end architecture;

```

**Explanation:
**

- We define the architecture `Behavioral` for our `multiplier_4x4` entity.
- Inside the architecture, we create a process sensitive to changes in the input signals `A` and `B`.
- We declare a variable called `temp`, which will hold the temporary results during the multiplication process. It is initialized to all zeros.
- Using nested loops, we iterate through each bit of `A` and `B`.
- If both the current bits of `A` and `B` are '1', it indicates a contribution to the product at position `i+j`. We set the corresponding bit in the `temp` variable to '1'.
- Once all the products are calculated and stored in `temp`, we assign the value of `temp` to the output port `Result`, representing the final 8-bit product.

## Conclusion

We hope this guide has been helpful in understanding how to write a 4-bit by 4-bit multiplier in VHDL. This multiplier can be seamlessly incorporated into various FPGA and ASIC projects for performing multiplication operations on 4-bit numbers. Before deploying the design in hardware, we recommend thoroughly simulating and testing it to ensure its functionality. If you have any questions or require further assistance, please do not hesitate to contact us. Happy coding and designing!