OFDM

DFT DFT

Inverse DFT

An N-point DFT (or inverse DFT) requires a total of N2 complex multiplications This transform can be implemented very efficiently by the fast Fourier

transform (FFT)

Serial-to-parallel

converter

Parallel-to-serial converter

Signal mapper

Signal demapper

IDFT

DFT

P/S

S/P

Guard interval /

cyclic prefix insertion

CP removal

channel

1,,1,0,2exp][)(1

0

NkN

nkjnxkX

N

n

1,,1,0,2exp][][1

0

NnN

nkjkXnx

N

n

OFDM OFDM implementation

An OFDM signal consists of a sum of subcarriers that are modulated by using QAM, and is implemented using the inverse DFT (discrete Fourier transform)

di : the i-th complex QAM symbol N : the number of subcarriers

1,,1,0,2exp)(1

0

NnN

injdns

N

ii

Serialto

parallelQAM data

dN-1 … d2 d1 d0

dN-1

d1

d0

X

X

X

Exp(j*2*pi*0*n/N)

Exp(j*2*pi*1*n/N)

Exp(j*2*pi*(N-1)*n/N)

+s(N-1) … s(2) s(1) s(0)

OFDM FFT

The FFT drastically reduces the amount of calculations by exploiting the regularity of the operations in the DFT

Using the radix-2 algorithm, an N-point FFT (or inverse FFT) requires only (N/2)log2N complex multiplications 256 multiplications for DFT versus 32 for FFT (for a 16-point transform) – a

reduction by a factor of 8 In VDSL system, 4096-point transform is used – a reduction by a factor of

about 683

Homework 16-point DFT & 16-point inverse DFT programming 16-point FFT & 16-point inverse FFT programming

(Tip) Definition for complex variable should be needed

Complex variable (real + imaginary, A=a+ib) Complex multiplication (A*B=(a+ib)*(c+id)=(ac-bd)+i*(ad+bc)) Complex addition (A+B=(a+ib)+(c+id)=(a+c)+i*(b+d)) Complex subtraction (A-B=(a-c)+i*(b-d))

Block diagram

마감 : 12 월 12 일 수요일 낮 12 시까지 ( 교수 이메일로 보낼것 ) 제출자료

source program (c-code) with comments .exe file (file_name = student_id.exe) for demonstration

Random generator

16 complex symbols Serial-to-

parallel

16-point inverse

DFT

16-point DFT

parallel-to-serial

printf(these values) printf(these values) printf(these values)

Random generator

16 complex symbols Serial-to-

parallel16-point

inverse FFT16-point

FFTparallel-to-

serial

printf(these values) printf(these values) printf(these values)