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
chadi.abourjeily@lau.edu.lb
September 29, 2011
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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.
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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.
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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)
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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.
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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)
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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)
]⟩
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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)
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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)
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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).
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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.
Chadi Abou-Rjeily Communication Systems Lecture-3: Orthogonal Representation of
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
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