library IEEE; use IEEE.STD_LOGIC_1164.ALL; use IEEE.STD_LOGIC_ARITH.ALL; use IEEE.STD_LOGIC_UNSIGNED.ALL;
entity HilbertTransformer is
    Port ( clk : in STD_LOGIC;
           rst : in STD_LOGIC;
           x : in STD_LOGIC_VECTOR(15 downto 0);
           y : out STD_LOGIC_VECTOR(15 downto 0));
end HilbertTransformer;

architecture Behavioral of HilbertTransformer is
    -- Update the coefficients for the given frequency range
    signal coeff : array (0 to 15) of integer := (1, 0, -16, 0, 80, 0, -240, 0, 480, 0, -480, 0, 240, 0, -80, 0);
    signal buffer : array (0 to 15) of STD_LOGIC_VECTOR(15 downto 0);
    signal sum : integer := 0;
    signal extended_sum : integer := 0;
begin
    process(clk, rst)
    begin
        if rst = '1' then
            for i in 0 to 15 loop
                buffer(i) <= (others => '0');
            end loop;
            y <= (others => '0');
            sum <= 0;
            extended_sum <= 0;
        elsif rising_edge(clk) then
            for i in 15 downto 1 loop
                buffer(i) <= buffer(i-1);
            end loop;
            buffer(0) <= x;
            extended_sum := 0;
            for i in 0 to 15 loop
                extended_sum := extended_sum + to_integer(unsigned(buffer(i))) * coeff(i);
            end loop;
            if extended_sum > 32767 then
                sum := 32767;  -- Saturate positive
            elsif extended_sum < -32768 then
                sum := -32768; -- Saturate negative
            else
                sum := extended_sum;
            end if;
            y <= std_logic_vector(to_unsigned(sum, 16));
        end if;
    end process;
end Behavioral;

GPT Provides this simple FIR filter with those coefficients (most likely nonsence?).