1536 FFT Implementation in LTE

 In an LTE downlink, the system needs support variable transmission bandwidths, such as 1.25 MHz, 2.5 MHz, 5 MHz, 10 MHz, 15 MHz, and 20 MHz. Each transmission bandwidth corresponds to a different fast Fourier transform (FFT) size of 128, 256, 512, 1024, 1536, and 2048 points. Table 1 lists the downlink transmission bandwidth, sampling frequency and FFT size. 

Transmission BW	1.4 MHz	2.5 MHz	5 MHz	10 MHz	15 MHz	20 MHz
Sampling Frequency	1.92 MHz	3.84 MHz	7.68 MHz	15.36 MHz	20.34 MHz	30.72 MHz
FFT size	128	256	512	1024	1536	2048

 Except 15 MHz bandwidth, all FFT sizes have 2^m form and can be implemented in the standard Cooley–Tukey algorithm using radix-2 FFT.  This essay discusses the implementation of 1536 FFT for 15 MHz bandwidth.

The FFT reduces the number of computations of DFT from O(N²) to O(NlogN) by decomposition in time or frequency domain. The DFT of a sequence x(n) is given by

In the current design, the transform length, N, is 1536. Using the decimation in time method, the first step is to divide the input sequence into three sequences. Each sequence goes through the 512-point FFT computation. The last step  combines the three sequences to get a final output result.

 

For N=1536, we have

 Therefore, a 1536-point FFT can be decomposed into three 512-point FFTs followed by a single radix-3 combinational stage. It should be noted that a FFT 512 has a period 512, if the number of frequency is larger than 512, we have

Where mod() denote the modular operation. The following flowchart describes the steps to calculate 1536 FFT for LTE downlink.

 

 

Following Matlab script implements the 1536 FFT.

 

N=1536;

N1 = 1536/3;

 

for i=1:N1

   

    y1(i) = x(3*i-2);

    y2(i) = x(3*i-1);

    y3(i) = x(3*i);

end

 

FFT_y1 = fft(y1,512);

FFT_y2 = fft(y2,512);

FFT_y3 = fft(y3,512);

 

for n=1:N

   

    FFT_y(n) = FFT_y1(mod((n-1),512)+1)+FFT_y2(mod((n-1),512)+1)*exp(-j*2*pi*(n-1)/1536);

    FFT_y(n) = FFT_y3(mod((n-1),512)+1)*exp(-j*2*2*pi*(n-1)/1536)+FFT_y(n);

End