Fourier Series Facilitates Representation of Any Periodic Signal in Terms of Sinusoids

FOURIER SERIES:

Fourier series facilitates representation of any periodic signal in terms of sinusoids.

∞

x(t)=A0+∑ Ak cos(kw0 t+θk)

k=1

Periodic signal x(t) is expressed as a sum of sinusoids of frequencies 0, w0, 2w0,… , Kw0,…

whose amplitudes are A0 ,A1 , A2,………… Ak,…..and whose phases are 0,θ1, θ2 ,………, θk,………

Plot of Ak Vs w gives amplitude spectrum and plot of θk Vs w gives phase spectrum.These two plots together are frequency spectra of x(t).

fig 1

fig 2

Example :for the time domain description of x(t) in fig 1 ,the frequency domain description of x(t) is given in fig 2.

REAL VALUED SIGNAL:

It is signal represented by real sinusoids.

∞

x(t)=A0+∑ Ak cos(kw0 t+θk)

k=1

Here A0, Ak and cosine components all are real.

COMPLEX EXPONENTIAL FOURIER SERIES:

A more general fourier series is complex exponential fourier series:

∞

x(t)= ∑ ak e jk w0 t

k= -∞

where, ak , e jk w0 t =complex numbers

HARMONICALLY RELATED COMPLEX EXPONENTIALS:

For above equation ,set of harmonically related complex exponential is associated with periodic signal x(t).

{ e j w0 t , e j2 w0 t ,…………}

For ak e jk w0 t

ak = ‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌ |ak|‌‌ <θk

= ‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌ |ak|‌‌ e j θk

ak e jk w0 t =‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌ |ak|‌‌ e j(k w0 t +θk)

=‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌ |ak|‌‌ cos(k w0+ θk) +j |ak|‌‌ sin(k w0+ θk)

For

A0 +∑ Ak cos(kw0 t+θk) ↔ ∑ ak e jk w0 t

We know, (Euler identity)

cos(kw0 t) = e jk w0 t + e -jk w0 t

2

sin (kw0 t) = e jk w0 t + e -jk w0 t

2j

‌ Ak cos(kw0 t)= (Ak/2) * (e jk w0 t + e -jk w0 t)

Ak cos(kw0 t +θk)= (Ak/2) * (e j(k w0 t +θk)+ e -j(k w0 t +θk))

=((Ak/2)* e j θk) e jk w0 t +((Ak/2)* e -j θk) e -jk w0 t

so,

Ak cos(kw0 t +θk) ↔ ak e jk w0 t + a-k e -jk w0 t

Where, ak and a-k are complex conjugates

|ak|‌‌ = Ak/2 =|a-k|‌‌

and < ak = θk = -< a-k

Fig 3

Fig 4

For fig 3 (real valued spectrum) the complex exponential frequency spectrum is in fig 4

[ Note: frequency of exponentials e ±j w0 t is |w0|‌‌ .Existence of the spectrum at negative frequency is indication of the fact that an exponential component e -jk w0 t exixts in the series.]

Similarly,

Ak sin(kw0 t +θk)= ((Ak/2j)* e j θk) e jk w0 t - ((Ak/2j)* e -j θk) e -jk w0 t

Where,

|ak|‌‌ = Ak/2 =|a-k|‌‌

and < ak = (θk – 90)= -< a-k

Now a general definition of fourier series of a periodic signal x(t) is

∞

x(t)= ∑ ak e jk w0 t

k= -∞

where, w0 =2 Π /T

T =fundamental time period