design of fft processor

.. Report by Rohit Singh and Amit Kumar Singh for Self –Study-Course, M.Tech. (VST),

Deptt. Of EE,SOE, Shiv Nadar University

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IMPLEMENTATION OF FAST FOURIER

TRANSFORM

Self Study Course

By

Rohit Singh

Amit Kumar Singh

(M.Tech (VST))



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Contents

1. Introduction to fast fourier transform ...………………… 03

2. Butterfly structures for FFT ...………………… 08

3. Implementation of 16-point FFT blocks …………………... 11

4. Matlab code for DFT …………………. 12

5. Matlab code for 8 point FFT …………………. 13

6. VHDL code for 16 point FFT Processor …………… 14



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1. INTRODUCTION TO FAST FOURIER TRANSFORM

A Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete

Fourier Transform (DFT) and its inverse. There are many distinct FFT algorithms

involving a wide range of mathematics, from simple complex-number arithmetic to group

theory and number theory. The fast Fourier Transform is a highly efficient procedure for

computing the DFT of a finite series and requires less number of computations than that

of direct evaluation of DFT. It reduces the computations by taking advantage of the fact

that the calculation of the coefficients of the DFT can be carried out iteratively. Due to

this, FFT computation technique is used in digital spectral analysis, filter simulation,

autocorrelation and pattern recognition.

The FFT is based on decomposition and breaking the transform into smaller

transforms and combining them to get the total transform. FFT reduces the computation

time required to compute a discrete Fourier transform and improves the performance by a

factor of 100 or more over direct evaluation of the DFT.

A DFT decomposes a sequence of values into components of different

frequencies. This operation is useful in many fields but computing it directly from the

definition is often too slow to be practical. An FFT is a way to compute the same result

more quickly: computing a DFT of N points in the obvious way, using the definition,

takes O( N2 ) arithmetical operations, while an FFT can compute the same result in only

O(N log N) operations.

The difference in speed can be substantial, especially for long data sets where N

may be in the thousands or millions—in practice, the computation time can be reduced by

several orders of magnitude in such cases, and the improvement is roughly proportional

to N /log (N). This huge improvement made many DFT-based algorithms practical. FFT’s

are of great importance to a wide variety of applications, from digital signal processing



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and solving partial differential equations to algorithms for quick multiplication of large

integers.

The most well known FFT algorithms depend upon the factorization of N, but

there are FFT with O (N log N) complexity for all N, even for prime N. Many FFT

algorithms only depend on the fact that is an N th primitive root of unity, and thus can be

applied to analogous transforms over any finite field, such as number-theoretic

transforms.

The Fast Fourier Transform algorithm exploit the two basic properties of the

twiddle factor - the symmetry property and periodicity property which reduces the

number of complex multiplications required to perform DFT.

FFT algorithms are based on the fundamental principle of decomposing the

computation of discrete Fourier Transform of a sequence of length N into successively

smaller discrete Fourier transforms. There are basically two classes of FFT algorithms.

A) Decimation In Time (DIT) algorithm

B) Decimation In Frequency (DIF) algorithm.

In decimation-in-time, the sequence for which we need the DFT is successively

divided into smaller sequences and the DFTs of these subsequences are combined in a

certain pattern to obtain the required DFT of the entire sequence. In the decimation-in-

frequency approach, the frequency samples of the DFT are decomposed into smaller and

smaller subsequences in a similar manner.

The number of complex multiplication and addition operations required by the

simple forms both the Discrete Fourier Transform (DFT) and Inverse Discrete Fourier

Transform (IDFT) is of order N2 as there are N data points to calculate, each of which

requires N complex arithmetic operations.

The discrete Fourier transform (DFT) is defined by the formula:



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;

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0)()(

NnKj

N

nenxKX

Where K is an integer ranging from 0 to N − 1.

The algorithmic complexity of DFT will O(N2) and hence is not a very efficient

method. If we can't do any better than this then the DFT will not be very useful for the

majority of practical DSP application. However, there are a number of different 'Fast

Fourier Transform' (FFT) algorithms that enable the calculation the Fourier transform of

a signal much faster than a DFT. As the name suggests, FFTs are algorithms for quick

calculation of discrete Fourier transform of a data vector. The FFT is a DFT algorithm

which reduces the number of computations needed for N points from O(N2) to O(N log N)

where log is the base-2 logarithm. If the function to be transformed is not harmonically

related to the sampling frequency, the response of an FFT looks like a ‘sinc’ function (sin

x) / x.

The Radix-2 DIT algorithm rearranges the DFT of the function xn into two parts:

a sum over the even-numbered indices n = 2m and a sum over the odd-numbered indices

n = 2m + 1:



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One can factor a common multiplier out of the second sum in the

equation. It is the two sums are the DFT of the even-indexed part x2m and the DFT of

odd-indexed part x2m + 1 of the function xn. Denote the DFT of the Even-indexed inputs

x2m by Ek and the DFT of the Odd-indexed inputs x2m + 1 by Ok and we obtain:

However, these smaller DFTs have a length of N/2, so we need compute only N/2

outputs: thanks to the periodicity properties of the DFT, the outputs for N/2 < k < N from

a DFT of length N/2 are identical to the outputs for 0< k < N/2. That is, Ek + N / 2 = Ek

and Ok + N / 2 = Ok. The phase factor exp[ − 2πik / N] called a twiddle factor which

obeys the relation: exp[ − 2πi(k + N / 2) / N] = e − πiexp[ − 2πik / N] = − exp[ − 2πik / N],

flipping the sign of the Ok + N / 2 terms. Thus, the whole DFT can be calculated as

follows:

This result, expressing the DFT of length N recursively in terms of two DFTs of

size N/2, is the core of the radix-2 DIT fast Fourier transform. The algorithm gains its

speed by re-using the results of intermediate computations to compute multiple DFT

outputs. Note that final outputs are obtained by a +/− combination of Ek and Okexp( −

2πik / N), which is simply a size-2 DFT; when this is generalized to larger radices below,

the size-2 DFT is replaced by a larger DFT (which itself can be evaluated with an FFT).



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This process is an example of the general technique of divide and conquers algorithms. In

many traditional implementations, however, the explicit recursion is avoided, and instead

one traverses the computational tree in breadth-first fashion.

Fig 1.1 Decimation In Time FFT

In the DIT algorithm, the twiddle multiplication is performed before the butterfly

stage whereas for the DIF algorithm, the twiddle multiplication comes after the Butterfly

stage. Fig 1.2 Decimation In Frequency FFT

The 'Radix 2' algorithms are useful if N is a regular power of 2 (N=2p). If we

assume that algorithmic complexity provides a direct measure of execution time and that

the relevant logarithm base is 2 then as shown in table 1.1, ratio of execution times for

the (DFT) vs. (Radix 2 FFT) increases tremendously with increase in N.



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The term 'FFT' is actually slightly ambiguous, because there are several

commonly used 'FFT' algorithms. There are two different Radix 2 algorithms, the so-

called 'Decimation in Time' (DIT) and 'Decimation in Frequency' (DIF) algorithms. Both

of these rely on the recursive decomposition of an N point transform into 2 (N/2) point

transforms.

Number

of Points,

N

Complex Multiplications

in Direct computations,

N2

Complex Multiplication

in FFT Algorithm, (N/2)

log2 N

Speed

improvement

Factor

4 16 4 4.0

8 64 12 5.3

16 256 32 8.0

32 1024 80 12.8

64 4096 192 21.3

128 16384 448 36.6

Table 1.1: Comparison of Execution Times, DFT & Radix – 2 FFT

2. BUTTERFLY STRUCTURES FOR FFT

Basically FFT algorithms are developed by means of divide and conquer method,

the is depending on the decomposition of an N point DFT in to smaller DFT’s. If N is

factored as N = r1,r2,r3 ..rL where r1=r2=…=rL=r, then rL =N. where r is called as Radix of

FFT algorithm.

If r= 2, then it is called as radix-2 FFT algorithm,. The basic DFT is of size of 2.

The N point DFT is decimated into 2 point DFT by two ways such as Decimation In

Time (DIT) and Decimation In Frequency (DIF) algorithm. Both the algorithm take the

advantage of periodicity and symmetry property of the twiddle factor.

NnKj

enKN

W

2



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The radix-2 decimation-in-frequency FFT is an important algorithm obtained by

the divide and conquers approach. The Fig. 1.3 below shows the first stage of the 8-point

DIF algorithm.

Fig. 1.3 First Stage of 8 point Decimation in Frequency Algorithm.

The decimation, however, causes shuffling in data. The entire process involves v

= log2 N stages of decimation, where each stage involves N/2 butterflies of the type

shown in the Fig. 1.3.

Fig. 1.4: Butterfly Scheme.



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Here N

nkj

eknW

2 is the Twiddle factor.

Consequently, the computation of N-point DFT via this algorithm requires (N/2)

log2 N complex multiplications. For illustrative purposes, the eight-point decimation-in

frequency algorithm is shown in the Figure below. We observe, as previously stated, that

the output sequence occurs in bit-reversed order with respect to the input. Furthermore, if

we abandon the requirement that the computations occur in place, it is also possible to

have both the input and output in normal order. The 8 point Decimation In frequency

algorithm is shown in Fig 1.5.

Fig. 1.5 8 point Decimation in Frequency Algorithm



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3. IMPLEMENTATION OF 16-POINT FFT BLOCKS

The FFT computation is accomplished in three stages. The x(0) until x(15)

variables are denoted as the input values for FFT computation and X(0) until X(15) are

denoted as the outputs. The pipeline architecture of the 16 point FFT is shown in Fig 2.1

consisting of butterfly schemes in it. There are two operations to complete the

computation in each stage.

Fig 2.1: Architecture of 16 point FFT.

The upward arrow will execute addition operation while downward arrow will

execute subtraction operation. The subtracted value is multiplied with twiddle factor



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value before being processed into the next stage. This operation is done concurrently and

is known as butterfly process.

4. Matlab Code for DFT

% program to calculate DFT.

clear all;

clc;

x=input ('enter the sequence ');

tic

N=size(x,2);

w=zeros(N);

for a=1:N

for b=1:N

w(a,b)=exp((-1j*2*pi*(a-1)*(b-1))/N);

end

end

Xk = w*x';

Xk

Toc



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5. Matlab Code for 8 point FFT

% program to calculate 8 point FFT clear all; clc; x=input (' enter the sequence of length 8 '); tic s0=zeros(1,8); s1=zeros(1,8); s2=zeros(1,8); s3=zeros(1,8); N=size(x,2); if (N < 8 || N > 8) disp(' sequence is of not proper length '); else stage1=[1 1 1 0-1j 1 1 1 0-1j]; stage2=[1 1 1 1 1 0.707-0.707j 0-1j -0.707-0.707j]; p=[0:7]'; t=bitrevorder(p); % calculation of stage zero for i=1:N s0(i)=x(t(i)+1); end % calculation of stage one for i=1:2:N s1(i)=s0(i)+s0(i+1); s1(i+1)=s0(i)-s0(i+1); end for a=1:N s1(a)=s1(a)*stage1(a); end % calculation of stage two for i=1:4:N s2(i)=s1(i)+s1(i+2); s2(i+1)=s1(i+1)+s1(i+3); s2(i+2)=s1(i)-s1(i+2); s2(i+3)=s1(i+1)-s1(i+3); end for a=1:N s2(a)=s2(a)*stage2(a); end % calculation of stage three s3(1)=s2(1)+s2(5); s3(2)=s2(2)+s2(6); s3(3)=s2(3)+s2(7); s3(4)=s2(4)+s2(8); s3(5)=s2(1)-s2(5); s3(6)=s2(2)-s2(6); s3(7)=s2(3)-s2(7); s3(8)=s2(4)-s2(8); end disp(' FFT result '); s3 toc



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6. VHDL CODE FOR 16 POINT FFT PROCESSOR

FFT MAIN PROGRAM

library IEEE;

use IEEE.STD_LOGIC_1164.ALL;

use IEEE.STD_LOGIC_SIGNED.all;

use IEEE.STD_LOGIC_ARITH.all;

use work.fft_package.all;

entity Parallel_FFT is

Port ( x : in signed_vector;

y : out comp_array);

end Parallel_FFT;

architecture Behavioral of Parallel_FFT is

component real_input_to_complex is

Port ( input_real : in signed_vector;

input_comp : out comp_array);

end component;

component butterfly is

port(

c1,c2 : in complex; --inputs

w :in complex; -- phase factor

g1,g2 :out complex -- outputs



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);

end component;

constant w : comp_array := (

("0100000000000000", "0000000000000000"),

("0011101100100001", "1110011110000010"),

("0010110101000001", "1101001010111111"),

("0001100001111110", "1100010011011111"),

("0000000000000000", "1100000000000000"),

("1110011110000010", "1100010011011111"),

("1101001010111111", "1101001010111111"),

("1100010011011111", "1110011110000010"),

("1100000000000000", "0000000000000000"),

("1100010011011111", "0001100001111110"),

("1101001010111111", "0010110101000001"),

("1110011110000010", "0011101100100001"),

("0000000000000000", "0100000000000000"),

("0001100001111110", "0011101100100001"),

("0010110101000001", "0010110101000001"),

("0011101100100001", "0001100001111110")

);

signal s, g1, g2, g3 : comp_array := (others =>

("0000000000000000","0000000000000000"));



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begin

--Real Input value transformed in complex value

sixteen_bits_complex_transform : real_input_to_complex port

map(x, s);

--first stage of butterfly's.

bf11 : butterfly port map(s(0),s(8),w(0),g1(0),g1(8));








--second stage of butterfly's.

bf21 : butterfly port map(g1(0),g1(4),w(0),g2(0),g2(4));








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bf27 : butterfly port

map(g1(10),g1(14),w(4),g2(10),g2(14));


map(g1(11),g1(15),w(6),g2(11),g2(15));

--third stage of butterfly's.








map(g2(12),g2(14),w(0),g3(12),g3(14));


map(g2(13),g2(15),w(4),g3(13),g3(15));

--fourth stage of butterfly's.

bf41 : butterfly port map(g3(0),g3(1),w(0),y(0),y(8));









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end Behavioral;

BUTTERFLY.VHD

library IEEE;



entity butterfly is

port(

c1,c2 : in complex; --inputs

w :in complex; -- phase factor

g1,g2 :out complex -- outputs

);

end butterfly;

architecture Behavioral of butterfly is

begin

--butterfly equations.

g1 <= add(c1,c2);



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g2 <= mult(sub(c1,c2),w);

end Behavioral;

FFT PACKAGE

library IEEE;




package fft_package is

type complex is

record

r : signed(15 downto 0);

i : signed(15 downto 0);

end record;

type signed_vector is array (0 to 15) of signed(7 downto

0);

type comp_array is array (0 to 15) of complex;

function add (n1,n2 : complex) return complex;

function sub (n1,n2 : complex) return complex;

function mult (n1,n2 : complex) return complex;



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end fft_package;

package body fft_package is

--addition of complex numbers

function add (n1,n2 : complex) return complex is

variable sum : complex;

begin

sum.r:=n1.r + n2.r;

sum.i:=n1.i + n2.i;

return sum;

end add;

--subtraction of complex numbers.

function sub (n1,n2 : complex) return complex is

variable diff : complex;

begin

diff.r:=n1.r - n2.r;

diff.i:=n1.i - n2.i;

return diff;

end sub;

--multiplication of complex numbers.

function mult (n1,n2 : complex) return complex is



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variable prod : complex;

variable prod_aux_r : signed(31 downto 0);

variable prod_aux_i : signed(31 downto 0);

begin

prod_aux_r := (n1.r * n2.r) - (n1.i * n2.i);

prod.r:= prod_aux_r(29 downto 14);

prod_aux_i := (n1.r * n2.i) + (n1.i * n2.r);

prod.i:= prod_aux_i(29 downto 14);

return prod;

end mult;

end fft_package;

REAL TO COMPLEX

library IEEE;





entity real_input_to_complex is



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Port ( input_real : in signed_vector;

input_comp : out comp_array);

end real_input_to_complex;

architecture Behavioral of real_input_to_complex is

component sig_ext_8_to_16_bits is

Port ( sig_in : in signed(7 downto 0);

sig_out : out complex);

end component;

begin

input0 : sig_ext_8_to_16_bits port map (input_real(0),

input_comp(0));


input_comp(1));


input_comp(2));


input_comp(3));


input_comp(4));


input_comp(5));


input_comp(6));


input_comp(7));



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input_comp(8));


input_comp(9));


input_comp(10));


input_comp(11));


input_comp(12));


input_comp(13));


input_comp(14));


input_comp(15));

end Behavioral;

EIGHT TO SIXTEEN

library IEEE;






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entity sig_ext_8_to_16_bits is

Port ( sig_in : in signed(7 downto 0);

sig_out : out complex);

end sig_ext_8_to_16_bits;

architecture Behavioral of sig_ext_8_to_16_bits is

signal sig_out_real : signed(15 downto 0);

begin

sig_out_real <= X"00" & sig_in when sig_in(7)='0' else

X"ff" & sig_in ;

sig_out <= (sig_out_real, "0000000000000000");

end Behavioral;

TEST BENCH

library IEEE;





ENTITY fft_cooley_tukey_tb IS

END fft_cooley_tukey_tb;



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ARCHITECTURE behavior OF fft_cooley_tukey_tb IS

-- Component Declaration for the Unit Under Test (UUT)

COMPONENT Parallel_FFT

Port ( x : in signed_vector;

y : out comp_array);

END COMPONENT;

--Inputs

signal x : signed_vector:= (others => "00000000");

--Outputs

signal y : comp_array;

BEGIN

-- Instantiate the Unit Under Test (UUT)

uut: Parallel_FFT PORT MAP (

x => x,

y => y

);

-- Stimulus process

stim_proc: process



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begin

-- hold reset state for 100 ns.

wait for 100 ns;

-- x<=("00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001"

-- );

-- x<=("00000000",



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-- "00000000",

-- "00000000",

-- "00000000",

-- "00000000",

-- "00000000",

-- "00000000",

-- "00000000",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001",

-- "00000001"

-- );

x<=("00000101",

"00001010",

"00001111",

"00010100",

"00011001",

"00011110",



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"00100011",

"00101000",

"00101101",

"00110010",

"00110111",

"00111100",

"01000001",

"01000110",

"01001011",

"01010000"

);

wait;

end process;

END;

design of fft processor

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