A new radix 4 fft algorithm

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 3, April (2013), © IAEME
251
A NEW RADIX-4 FFT ALGORITHM
Dr. Syed Abdul Sattar1
, Dr. Mohammed Yousuf Khan2
, and Shaik Qadeer3
1
(Professor & Dean. RITS Chevella. R R Dist. A P . India)
2
(Principal Polytechnic, MANUU, Hyderabad, India)
3
(Department of EED, MJCET, Hyderabad, India)
ABSTRACT
The Radix-4 Fast Fourier Transform (FFT) is widely accepted for signal processing
applications in wireless communication system. Here, we present a new Radix-4 FFT which reduces
the operational count by 6% lesser than standard Radix-4 FFT without losing any arithmetic accuracy.
Simulation results are also given for the verification of the algorithm.
Keywords: Discrete Fourier Transform (DFT); Decimation in Time (DI|T); Fast Fourier Transform
(FFT); and Flop counts.
I. INTRODUCTION
The Discrete Fourier transform (DFT) is among the most fundamental operation in digital
signal processing applications [1, 2, 7]. The algorithm to compute DFT is called as FFT. When
considering the implementations, the FFT/IFFT algorithm should be chosen keeping in view the
execution speed, hardware complexity, flexibility and precision [8, 9]. Most of the above mentioned
parameters depend on the exact count of arithmetic operations (real additions and multiplications),
herein called flop counts, required for a DFT of a given size N which remains an intriguing unsolved
mathematical question.
There are various types of FFT algorithms namely Radix-2/4/8 and Split radix including DIT
and DIF versions. The flop counts of standard Radix-4 DIT is [5]:
NNNT 2log
4
1
4)( =
(1)
Here we propose a new Radix-4 DIT algorithm which computes FFT in flop counts:
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN
ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 3, April 2013, pp. 251-256
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
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© I A E M E

252
NNNT 2log
8
1
4)( =
(2)
Saving of flop counts is almost 6% as compared to the standard Radix-4 FFT. In this paper a
simple recursive method is adopted to get the saving. The rest of the paper is organized as follows:
Section II covers the new algorithm principle, in section III operation count with the implementation
of proposed algorithm is discussed and conclusion in section IV.
II. NEW RADIX-4 DIT ALGORITHM
In this section we suggest a modification to standard Radix-4 FFT algorithm to reduce the
number of multiplications. To derive the algorithm, recall that the DFT is defined by [5]
1..0,
1
0
−=∑=
−
=
NnWxX kn
N
N
k
nk (3)
where WN
K
= exp (-j2π/N). Then for N divisible by 4, we perform a decimation in time
decomposition of xn into four smaller DFTs, of x4n , x4n+2(the even elements), x4n+1 and
x4n-1 (where x-1= xN-1)- this last sub-sequence would be in x4n+3 standard Radix-4 FFT, but here is
shifted cyclically by -4 [4]. We obtain:
,4/
14/
0
144/
14/
0
144/
14/
0
24
2
4/
14/
0
4 WxWWxWWxWWxX kn
N
N
k
n
k
N
kn
N
N
k
n
k
N
kn
N
N
k
n
k
N
kn
N
N
k
nk ∑+∑+∑+∑=
−
=
−
−
−
=
+
−
=
+
−
=
(4)
Where the Wk
N and W-k
N are the conjugate pair of twiddle factor. The four recurrent results
with subtransforms denoted by Yk, Gk , Hk, and Zk are shown below:
( )
( )
( )HWWZ kjGWYX
andHWWZ kjGWYX
HWWZ kGWYX
HWWZ kGWYX
k
k
N
k
Nk
k
NkNk
k
k
N
k
Nk
k
NkNk
k
k
N
k
Nk
k
NkNk
k
k
N
k
Nk
k
Nkk
−
+
−
+
−
+
−
++−=
−−−=
+−+=
+++=
2
4/3
2
4/
2
2/
2
,
,
,
(5)
The algorithm for this is shown in figure 1. The total flop count for this is given in (1).
Observe that it is equal to standard radix-4 FFT [5]. The key for the new algorithm is observation that
algorithm 1, both Hk, and Zk are multiplied by a same trigonometric function (Wk
N and W-k
N ).
Therefore to get saving we can divide Hk, and Zk by a common factor and multiply with the same in
the next iteration. Scaling factor ( Wavelet) proposed in [4] can be used as factor for getting saving. It
is having the following properties for k4=k mod N/4:

253
) ( ) ( )
( ) ( )
)
)
)
S
S
WtLet
SS
skremainingtheforandkofvalueshalffirstfor
S
S
otherwiseSNk
NkforSNk
forN
SDef
kN
kNk
NkN
kNNkN
kN
kN
kN
kNkN
,
,4/
,
,4/,
,4/
,
4,4/
4,4/,
4
3
,'cossin2
/42sin
8/4/42cos
41
:1
=
=
=





≤
≤
=
+
θθ
π
π
(6)
We use tN,k in our new algorithm which is always tan1 j±± or j±± cot . Computation of tN,k
takes 4 flops, whereas WN
k
takes 6. The new algorithm is shown in figure 2. Here in this new
algorithm , we first multiply by SN/4,k in algo1 to both Hk, and Zk which does not take any additional
multiplications, however it help us in getting saving for multiplications in algo2 of the new
algorithm.
Figure:1 Std. Radix-4 FFT algorithm

254
Figure:2 New . Radix-4 FFT algorithm
III. COMPUTATIONAL COUNT
The algorithms shown in figure 1 and figure 2 have same number of additions however due to
scaling and recalling operation in algorithm 2-3 we can save multiplications. In algo1 as the numbers
of additions as well as multiplications remain same, so we get





=
≥=+





+





=
4,16
44,17
4
22
4
12
)(1 2
Nif
Nif
N
T
N
T
NT
nN
(7)

255
In algo2 we save 4 real multiplications but lost 2. So the net saving is 2 real multiplications in
each iterations,





=
≥=+





+





=
4,16
44,5.16
4
22
4
12
)(2 2
Nif
Nif
N
T
N
T
NT
nN
(8)
Solving (7) and (8) by standard generating function method [3, 6] we get the total flop count
T(N) as given in (2). Hence a net saving of 6% is achieved.
Figure:3: Comparative Magnitude plot for N=4096.
Figure: 4: Comparative angle plot for N=4096.

256
IV. SIMULATION RESULTS
To validate the algorithm a computer simulation using Matlap for N=4096 is performed as
shown in figure 3 and figure 4. The results are matching with standard Radix -4 FFT algorithm
results. Figure 3 shows a comparative plot of magnitude between proposed FFT and standard FFT
whereas figure 4 covers for angle.
V. CONCLUSION
In this paper a new Radix-4 FFT algorithm is proposed. It is shown that the proposed FFT
algorithm is computational efficient without losing any accuracy. The proposed algorithm is suitable
for ASIC implementation as it is symmetric, unlike split radix FFT.
REFERENCES
[1] W.-H. Chang and T. Nguyen, On the Fixed-Point Accuracy Analysis of FFT Algorithms, IEEE
Transactions On Signal Processing, 56( 10), 2008, 4673-4682.
[2] Wade Lowdermilk and Fred Harris, Finite Arithmetic Considerations for the FFT
implemented in FPGA-Based Embedded Processors in Synthetic Instruments, IEEE
Instrumentation and Measurement Magazine, August 2007.
[3] D. E. Knuth, Fundamental Algorithms, 3rd ed., ser. The Art of Computer Programming
(Addison-Wesley, 1997, vol. 1.)
[4] ] M. Frigo and S. G. Johnson, A modified Split Radix FFT with fewer arithmetic operations,
IEEE Trans. Signal processing 55 (1), 111–119 (2007).
[5] Eleanor Chu and Alan George, Inside the FFT Black Box: Serial and Parallel FFT algorithms,
(CRC Press) Page No.282, Appendix A.
[6] D. E Knuth, Seminumerical Algorithms, 3rd ed., ser. The Art of Computer Programming
(Addison-Wesley, 1998, vol. 2.)
[7] M.Z.A. Khan and Shaik Qadeer, Streamlined Real Factor EUSIPCO2010, Aalborg, Denmark,
August23-27,2010
[8] A. V. Oppenheim and R. Schafer, Digital Signal Processing, .Pearson Education, 2004.
[9] J.G. Proakis and D.G. Manolakis,, Digital Signal Processing Principles, Algorithms, and
applications ,Pearson 3rd
Edition
[10] Abhishek choubey, Mayuri Kulshreshtha and Karunesh, “Determination of Optimum FFT for
Wi-Max under Different Fading”, International journal of Electronics and Communication
Engineering &Technology (IJECET), Volume 3, Issue 1, 2012, pp. 139 - 146, ISSN Print: 0976-
6464, ISSN Online: 0976 –6472.
[11] Kamatham Harikrishna and T. Rama Rao, “An Efficient Radix-22 FFT for Fixed & Mobile
Wimax Communication Systems”, International journal of Electronics and Communication
Engineering &Technology (IJECET), Volume 3, Issue 3, 2012, pp. 265 - 279, ISSN Print:
0976- 6464, ISSN Online: 0976 –6472.

A new radix 4 fft algorithm

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