Signals and Systems - Electrical Engineering

12.4 Application to Digital Communications 741

(wherep(t−mT)=1),

1

T

(m∫+ 1 )T

mT

r(t)e−j^2 πfktdt=

1

T

(m∫+ 1 )T

mT

N∑− 1

`= 0

d`ej^2 πf`te−j^2 πfktdt

=

N∑− 1

`= 0

d`

1

T

(m∫+ 1 )T

mT

e−j^2 π(fk−f`)tdt

=

N∑− 1

`= 0

d`δ[k−`]=dk

for any−∞<m<∞, and where we letfk−f`=(k−`)1f=(k−`)/T.

OFDM Implementation with FFT

If the modulated signals(t), 0≤t≤T, in Equation (12.27) is sampled att=nT/N, we obtain for a

frame the inverse DFT

s[n]=

N∑− 1

k= 0

dkej^2 πfknT/N=

N∑− 1

k= 0

dkej^2 πkn/N 0 ≤n≤N− 1 (12.28)

where 2πfkT/N= 2 πk/Nare the discrete frequencies in radians. At the receiver, with no interferences

present, the symbols{dk}are obtained by computing the DFT of the baseband received signal. Given

that the inverse and the direct DFT can be efficiently implemented by the FFT, the OFDM is a very effi-

cient technique that is used in wireless local area networks (WLANs) and digital audio broadcasting

(DAB).

Figure 12.14 gives a general description of the transmitter and receiver in an OFDM system: The high-

speed data in binary form coming into the system are transformed from serial to parallel and fed into

an IFFT block giving as output the transmitting signal that is sent to the channel. The received signal

is then fed into an FFT block providing estimates of the sent symbols that are finally put in serial

form.

FIGURE 12.14
Discrete model of baseband OFDM. The
blocks S/P and P/S convert a serial into a
parallel stream and a parallel to serial,
respectively.

S/P IFFT FFT P/S

d 0

d 1 Channe

{dk}

dN− 1

s(n) r(n)

d 0

∧

{dk}

∧

dN− 1

∧

d 1

∧

......