13.1 FOURIER TRANSFORMS
Ignoring in the present context the effect of the termAaexp(iωct), which gives a
contribution to the transmitted spectrum only atω=ωc, we obtain for the new
spectrum
̃g(ω)=1
√
2 πA∫∞−∞f(t)eiωcte−iωtdt=1
√
2 πA∫∞−∞f(t)e−i(ω−ωc)tdt=A ̃f(ω−ωc), (13.34)which is simply a shift of the whole spectrum by the carrier frequency. The use
of different carrier frequencies enables signals to be separated.
13.1.6 Odd and even functionsIff(t) is odd or even then we may derive alternative forms of Fourier’s inversion
theorem, which lead to the definition of different transform pairs. Let us first
consider an odd functionf(t)=−f(−t), whose Fourier transform is given by
̃f(ω)=√^1
2 π∫∞−∞f(t)e−iωtdt=1
√
2 π∫∞−∞f(t)(cosωt−isinωt)dt=− 2 i
√
2 π∫∞0f(t)sinωt dt,where in the last line we use the fact thatf(t)andsinωtare odd, whereas cosωt
is even.
We note that ̃f(−ω)=− ̃f(ω), i.e. ̃f(ω) is an odd function ofω. Hencef(t)=1
√
2 π∫∞−∞̃f(ω)eiωtdω=√^2 i
2 π∫∞0̃f(ω)sinωt dω=2
π∫∞0dωsinωt{∫∞0f(u)sinωu du}
.Thus we may define theFourier sine transform pairfor odd functions:
̃fs(ω)=√
2
π∫∞0f(t)sinωt dt, (13.35)f(t)=√
2
π∫∞0̃fs(ω)sinωt dω. (13.36)Note that although the Fourier sine transform pair was derived by considering
an odd functionf(t) defined over allt, the definitions (13.35) and (13.36) only
requiref(t)and ̃fs(ω) to be defined for positivetandωrespectively. For an