ELG3175 Introduction to Communication Systems Fourier transform, Parseval’s theoren, Autocorrelation and Spectral Densities Fourier Transform of a Periodic Signal • A periodic signal can be expressed as a complex exponential Fourier series. • If x(t) is a periodic signal with period T, then :

n ∞ j2π t T x(t) = ∑ X n e n=−∞ • Its Fourier Transform is given by:

n n  ∞ j2π t  ∞  j2π t  ∞ F T F T X ( f ) =  ∑ X ne  = ∑ X n e  = ∑ X nδ ()f − nf o n=−∞  n=−∞   n=−∞

Lecture 4 Example x(t )

A … …

0.25 0.5 0.75 t -A ∞ 2A ∞  2A  x(t) = ∑ e j4πnt = ∑ − j e j4πnt n=−∞ jπn n=−∞  πn  n is odd n is odd ∞  2A  X ( f ) = ∑ − j δ ()f − 2n n=−∞  πn  n is odd

Lecture 4 Example Continued

|X(f)|

2A/π 2A/π

2A/3 π 2A/3 π 2A/5 π 2A/5 π

-10 -6 -2 2 6 10 f

Lecture 4 Energy of a periodic signal

• If x(t) is periodic with period T, the energy on one period is:

| ( |) 2 E p = ∫ x t dt T

• The energy on N periods is EN = NE p. • The average normalized energy is E = lim NE p = ∞ N→∞

• Therefore periodic signals are never energy signals.

Lecture 4 Average normalized power of periodic signals • The power of x(t) on one period is : 1 P = | x(t |) 2 dt p T ∫ T • And it’s power on N periods is :

 iT  1 1  N 1  1 P = | x(t |) 2 dt = | x(t |) 2 dt = × NP = P Np NT ∫ N ∑ T ∫  N p p NT  i=1 (i− )1 T 

• It’s average normalized power is therefore P = lim PNp = Pp N →∞ • Therefore, for a periodic signal, its average normalized power is equal to the power over one period.

Lecture 4 Hermetian symmetry of Fourier Transform when x(t) is real

*  ∞  X * ( f ) =  ∫ x(t)e− j2πft dt  −∞  ∞ = ∫ x* (t)e j2πft dt −∞ ∞ = ∫ x(t)e j2πft dt −∞ ∞ = ∫ x(t)e− j2π (− f )t dt −∞ = X (− f )

Lecture 4 Parseval’s Theorem

• Let us assume that x(t) is an energy signal. • Its average normalized energy is : ∞ 2 E = ∫ x(t) dt −∞∞ = ∫ x(t)x* (t)dt −∞ ∞  ∞  j2πft * ∞ ∞ =  X ( f )e df x (t)dt 2 2 ∫ ∫  x(t) dt = X ( f ) df −∞−∞  ∫ ∫ ∞ ∞ −∞ −∞ = ∫X ( f ) ∫ x* (t)e − j2π (− f )t dt df −∞ −∞ ∞ ∞ 2 = ∫ X ( f )X * ( f )df = ∫ X ( f ) df −∞ −∞

Lecture 4 Example ∞ ∫sinc 2 (t)dt = ? −∞ ∞ ∞ ∫ sinc 2 (t)dt = ∫| sinc (t |) 2 dt −∞ −∞ ∞ = ∫| Π( f |) 2 df −∞ 2/1 = ∫ df − 2/1 = 1

Lecture 4 Autocorrelation function of energy signals • The autocorrelation function is a measure of similarity between a signal and itself delayed by τ. • The function is given by : ∞ * ϕx (τ ) = ∫ x(t)x (t −τ )dt −∞ • We note that ∞ ∞ )0( ( ) * ( ) ( ) 2 ϕ x = ∫ x t x t dt = ∫ x t dt = E −∞ −∞ * • We also note that ϕ x (τ ) = x(τ *) x (−τ )

Lecture 4 Energy spectral density

• Let x(t) be an energy signal and let us define G(f) = |X(f)|2 • The inverse Fourier transform of G(f) is ∞ ∞ ∫ G( f )e j2πfτ df = ∫ X ( f )X * ( f )e j2πfτ df −∞ −∞ ∞ ∞ = ∫ ∫ x(t)e− j2πft dtX * ( f )e j2πfτ df −∞−∞ ∞ ∞ = ∫x(t) ∫ X * ( f )e j2πfτ e− j2πft dfdt −∞ −∞ ∞ ∞ = ∫x(t) ∫ X * ( f )e j2πfτ e j2πf (−t)dfdt −∞ −∞ ∞ = ∫ x(t)x* (t −τ )dt −∞

Lecture 4 Energy spectral density

• The energy spectral density G(f) = F{ϕ(τ)}. • Example – Find the autocorrelation function of Π(t).

Lecture 4 Energy spectral density of output of LTI system

• If an energy signal with ESD Gx(f) is input to an LTI system with impulse response H(f), then the output,

y(t), is also an energy signal with ESD Gy(f) = 2 Gx(f)| H(f)| .

Lecture 4 ESD describes how the energy of the signal is distributed throughout the spectrum |H(f)| 2 |X(f)| 2 1

-f c fc f |Y(f)| 2

2 2 |X( fc)| Ey = 2| X(f)| ∆f ∆f is Hz, therefore |X(f)| 2 is in J/Hz

-f c fc f

Lecture 4 Autocorrelation Function of Power Signals

• We denote the autocorrelation of power signals as Rx(τ). • Keeping in mind that for the energy signal

autocorrelation, ϕ(0)= Ex, then for the power signal autocorrelation function Rx(0) should equal Px. • Therefore

T 1 * Rx (τ ) = lim x(t)x (t −τ )dt T →∞ ∫ 2T −T

Lecture 4 Power Spectral Density

• Power signals can be described by their PSD. • If x(t) is a power signal, then its PSD is denoted as F Sx(f), where Sx(f) = {Rx(τ)}. • The PSD describes how the signal’s power is distributed throughout its spectrum. • If x(t) is a power signal and is input to an LTI system, then the output, y(t), is also a power signal with PSD 2 Sy(f) = Sx(f)| H(f)| .

Lecture 4