Lecture03_Orthogonal Representation of Signals

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Communication Systems Lecture-3:Orthogonal Representation of Signals

Chadi Abou-Rjeily

Department of Electrical and Computer EngineeringLebanese American University

[email protected]

September 29, 2011

Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of

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Orthogonal Functions

The functions φn(t) and φm(t) are said to be orthogonal overthe interval [a b] if they satisfy the condition:

∫ b

a

φn(t)φ∗m(t)dt = 0 where n 6= m

Furthermore, if the functions in the set {φn(t)} areorthogonal, then:

∫ b

a

φn(t)φ∗m(t)dt =

{0, n 6= m;Kn, n = m.

= Knδnm

where:

δnm ,

{0, n 6= m;1, n = m.

The set {φn(t)} forms an orthonormal set when Kn = 1.This can be achieved by dividing φn(t) with

√Kn.

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Orthogonal Functions: Example (1)

Functions in the set {ejnω0t}n∈Z are orthogonal over the interval[a a + T0] ∀ a where ω0 = 2πf0 = 2π 1

T0.

Proof:

Replacing φn(t) = ejnω0t and φm(t) = ejmω0t :

∫ a+T0

a

φn(t)φ∗m(t)dt =

∫ a+T0

a

ejnω0te−jmω0tdt =

∫ a+T0

a

ej(n−m)ω0tdt

For n = m, ej(n−m)ω0t = 1 and:

∫ a+T0

a

φn(t)φ∗m(t)dt =

∫ a+T0

a

1dt = T0

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Orthogonal Functions: Example (2)

For n 6= m:∫ a+T0

a

φn(t)φ∗m(t)dt =

∫ a+T0

a

ej(n−m)ω0tdt

=ej(n−m)ω0(a+T0) − ej(n−m)ω0a

j(n − m)ω0

=ej(n−m)ω0a

[ej(n−m)ω0T0 − 1

]

j(n − m)ω0

=ej(n−m)ω0a

[ej(n−m)2π − 1

]

j(n − m)ω0

= 0where the last equation follows since: e j2πn = 1 for n ∈ Z.

Note that the functions in the set {ejnω0t}n∈Z are orthogonal(and not orthonormal since Kn = T0 6= 1).

On the other hand, functions in the set { 1√T0

ejnω0t}n∈Z are

orthonormal.

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Orthogonal Series (1)

Theorem: The waveform w(t) can be represented over theinterval [a b] by the series:

w(t) =∑

n

anφn(t)

where the orthogonal coefficients are given by:

an =1

Kn

∫ b

a

w(t)φ∗n(t)dt

Proof:

We need to show the existence of the coefficients {an} suchthat:

w(t) =∑

n

anφn(t)

Multiplying both sides of the previous equation by φ∗m(t):

w(t)φ∗m(t) =

[∑

n

anφn(t)

]

φ∗m(t)

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Orthogonal Series (2)

Integrating over the interval [a b]:

∫ b

a

w(t)φ∗m(t)dt =

∫ b

a

[∑

n

anφn(t)

]

φ∗m(t)dt

=∑

n

an

∫ b

a

φn(t)φ∗m(t)dt

=∑

n

anKnδnm

= amKm

Consequently:

am =1

Km

∫ b

a

w(t)φ∗m(t)dt

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Fourier Series

As a special case of the orthogonal series representation of signals,we have the Fourier series representation.

Consider the interval [a b] where b = a + T0. For the Fourierseries representation, the orthogonal functions are given by:

φn(t) = ejnω0t

where: ω0 = 2πf0 = 2π 1T0

= 2π 1b−a

.

In this case, any signal w(t) can be represented over [a b] by:

w(t) =∑

n∈Z

cnejnω0t

where the Fourier series coefficients are given by:

cn =1

T0

∫ a+T0

a

w(t)e−jnω0tdt

For mathematical convenience, a is chosen to take the valuea = 0 or a = −T0/2.

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Periodic waveforms

If the signal w(t) is periodic (with period T0), then theFourier series representation is valid over the entire interval−∞ < t < +∞.

Properties of the Fourier series:

If w(t) is real:cn = c∗

−n

If w(t) is real and even:

Im[cn] = 0

If w(t) is real and odd:

Re[cn] = 0

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Spectrum of Periodic waveforms

The spectrum of a periodic waveform w(t) having a period T0 isgiven by:

W (f ) =∑

n∈Z

cnδ(f − nf0)

where f0 = 1T0

and {cn} are the Fourier coefficients of w(t).

Proof:

w(t) can be expressed as (over the interval [−∞ + ∞]):

w(t) =∑

n∈Z

cnejnω0t

Given that the Fourier transform of the exponential functionej2πf0t is δ(f − f0), then the Fourier transform of w(t) takesthe following form:

W (f ) =∑

n∈Z

cnδ(f − nf0)

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Power Spectral Density of Periodic waveforms (1)

The PSD of a periodic waveform w(t) having a period T0 is givenby:

Pw (f ) =∑

n∈Z

|cn|2δ(f − nf0)

where f0 = 1T0

and {cn} are the Fourier coefficients of w(t).Proof:

We first start by calculating the autocorrelation function:

Rw (τ) = 〈w(t)w(t + τ)〉

Since w(t) is real, then w(t) = w∗(t) and:

Rw (τ) = 〈w∗(t)w(t + τ)〉

Replacing w(t) by its Fourier series representation results in:

Rw (τ) =

⟨[∑

n∈Z

c∗ne−jnωot

][∑

m∈Z

cmejmωo (t+τ)

]⟩

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Power Spectral Density of Periodic waveforms (2)

The above equation simplifies to:

Rw (τ) =

⟨∑

n∈Z

m∈Z

c∗ncmejmωoτej(m−n)ωo t

=∑

n∈Z

m∈Z

c∗ncmejmωoτ

ej(m−n)ωo t⟩

Given that:⟨ej(m−n)ωo t

⟩= δnm:

Rw (τ) =∑

n∈Z

|cn|2ejnω0τ

The PSD is equal to the Fourier transform of Rw (τ):

Pw (f ) = F [Rw (τ)] =∑

n∈Z

|cn|2F [ejnω0τ ]

=∑

n∈Z

|cn|2δ(f − nf0)

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Power Spectral Density of Periodic waveforms (3)

Consider the autocorrelation functionRw (τ) =

n∈Z|cn|2ejnω0τ . This function can be written as:

Rw (τ) =−1∑

n=−∞

|cn|2ejnω0τ + |c0|2 ++∞∑

n=1

|cn|2ejnω0τ

=

+∞∑

n=1

|c−n|2e−jnω0τ + |c0|2 +

+∞∑

n=1

|cn|2ejnω0τ

Since w(t) is real, then c−n = c∗

n implying that:

|c−n|2 = |c∗

n |2 = |cn|2

Consequently:

Rw (τ) = |c0|2 +

+∞∑

n=1

|cn|2[e−jnω0τ + ejnω0τ

]

= |c0|2 + 2

+∞∑

n=1

|cn|2 cos(nω0τ)

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Power Spectral Density of Periodic waveforms (4)

Note that (as expected from the general properties of theautocorrelation function):

Rw (τ) is real.Rw (τ) is even since the cosine terms {cos(nω0τ)}+∞

n=1 are even.Rw (τ) takes its maximum value at τ = 0 since the cosineterms {cos(nω0τ)}+∞

n=1 are maximum for τ = 0.The periods of the different terms in Rw (τ) are as follows:

Rw (τ) = |c0|2+2|c1|2 cos(ω0τ)︸ ︷︷ ︸

period: T0

+2|c2|2 cos(2ω0τ)︸ ︷︷ ︸

period: T0/2

+2|c3|2 cos(3ω0τ)︸ ︷︷ ︸

period: T0/3

+ · · ·

implying that the period of Rw (τ) is equal to T0 which is thesame as the period of w(t).

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Power Spectral Density of Periodic waveforms (5)

Consider the PSD Pw (f ) =∑

n∈Z|cn|2δ(f − nf0).

As expected from the general properties of the PSD:

Pw (f ) is real.Pw (f ) is non-negative.Pw (f ) is even since |c−n|2 = |c∗

n |2 = |cn|2 (note that the term|c−n|2 corresponds to the frequency −nf0 while the term |cn|2corresponds to the frequency nf0).

The average power of w(t) can be calculated from:

P = Rw (0) =∑

n∈Z

|cn|2

In the same way:

P =

∫ +∞

−∞Pw (f )df =

n∈Z

|cn|2

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Power Transfer function of LTI systems

Consider a linear and time-invariant system whose transfer function(frequency response) is H(f ) = F [h(t)] (where h(t) is the impulseresponse).

The input and output are related to each other by:

Y (f ) = H(f )X (f )

The PSDs of the input and output are related to each otherby:

Py (f ) = |H(f )|2Px(f )

, Gh(f )Px(f )

where Gh(f ) = |H(f )|2 is the power transfer function of theLTI system.

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Distortion-less Transmissions

In communication systems, a distortion-less channel is oftendesired.

A channel is distortion-less if its output is proportional to adelayed version of the input:

y(t) = Ax (t − Td)

where A is the gain and Td is the delay.

In the frequency domain, the last equation implies that:

Y (f ) = Ae−j2πTd f︸ ︷︷ ︸

H(f )

X (f )

In other words, two requirements must be satisfied:The amplitude response is flat:

|H(f )| = A = constant

The phase response is a linear function of frequency:

arg [H(f )] = −2πTd f

Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of