ee140 introduction to communication systems lecture...
TRANSCRIPT
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EE140 Introduction to
Communication Systems
Lecture 3
Instructor: Prof. Xiliang Luo
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Contents
• Deterministic signals
– Classification of signals
– Review of Fourier Transform
– Frequency-domain properties
– Time-domain properties
– Vector space and orthogonality
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Spectrum• The Fourier transform of a continuous-time signal is
a complex signal
• (Magnitude) spectrum
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)f(Sje)f(S)f(S
)f(S
Parseval’s Theorem• Parseval’s Theorem
– when , we have
• Energy Spectral Density
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df)f(S)f(Sdt)t(s)t(s **
2121
)t(s)t(s 21
df)f(Sdt)t(sE
22
Energy Energy Spectral Density
2)f(S)f(G Joules/Hz
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Example: Rectangular waveform• Rectangular waveform
• Frequency response
• Energy spectral density
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20
21
/t
/t)t(ga
τ
τ
)f(f
)fsin()ee(
fjdte)f(G fjfj/
/
ftja τπτ
τπ
τπτ
πτπτπ
τ
τ
π sinc 2
12
2
2
)f()f(G)f(G a τπτ 222sinc
1
(b) Ga(f)
t0
(a) ga(t)
Ga(f)ga(t)
Power Spectral Density (PSD)• Truncated waveform
• ESD
• Average normalized power
• Power spectral density
6
otherwise
TtTtstsT
,0
2/2/),()(
df)f(Sdt)t(sE TTT
22
21lim )f(S
T)f(P T
T
df)f(ST
dt)t(sT
dt)t(sT
P
TT
TT
/T
/TT
2
22
2
2
1lim
1lim
1lim
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Line Spectra for Periodic Signal• Periodic signal with period
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)t(s 0T
)nff(C
dteC
dteeC
dte)t(s)t(s)f(S
-nn
-n
t)nff(jn
ftj
-n
tnfjn
ftj
0
2
22
2
0
0
δ
π
ππ
πY
PSD of Periodic Signal
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• Periodic signal with period
• With Parseval’s Theorem
)t(s 0T
2
2
2
0
2
2
2 0
0
11lim
/T
/T
/T
/TTdt)t(s
Tdt)t(s
TP
)nff(Cdt)t(sT
P-n
n
/T
/T 0
22
2
2
0
0
0
1
δ
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Example• Spectrum of the following signal
• Fourier series
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0 T-Tt
V
s(t)
t),Tt(s)t(s
)/T(t/,
/t/,V)t(s
220
22
ττ
ττ
C
n
T
n
T
Vnfsin
Tnf
V
nfj
ee
T
V
enfj
V
TdtVe
TC
/nfj/nfj
/
/
tnfj/
/
tnfjn
πτττπ
ππ
π
τπτπ
τ
τ
πτ
τ
π
sinc2
2
11
000
2222
2
2
2
0
2
2
2
00
00
n n
tnfjtnfjn e
T
n
T
VeC)t(s 00 22 sinc ππ πττ
Example (cont’d)• PSD of the following signal
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0 T-Tt
V
s(t)
t),Tt(s)t(s
)/T(t/,
/t/,V)t(s
220
22
ττ
ττ
n
n
)nff(fT
V
)nff()f(C)f(P
02
2
0
2
sinc δπττ
δ
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Exercise: Question
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Exercise: Solution
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Properties of PSD Functions
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Exercise: Question
Exercise: Solution
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Contents
• Deterministic signals
– Classification of signals
– Review of Fourier Transform
– Frequency-domain properties
– Time-domain properties
– Vector space and orthogonality
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Autocorrelation Function• Autocorrelation function
• Autocorrelation function and the PSD are Fourier Transform pairs.
• Power of the signal
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2
2
1lim
/T
/T
*
T
*
dt)t(s)t(sT
)t(s)t(s)(R
τ
ττ
)(R)f(P τY
Pdt)t(sT
)(R/T
/TT
2
2
21lim0
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Example• Autocorrelation function of s(t) = Acos(t+)
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)ff(A
)ff(A
)nff()f(C)f(Pn
0
2
0
2
0
2
44
δδ
δ
τ
τ
ττ
τπ
cosA
]ee[A
dfe)f(P)(R
jj
fj
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22
2
Example• Determine the autocorrelation function of the
rectangular pulse waveform
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t
f(t)*f(t)A2T
2T0 T(c)
rf()=f()*f(-)A2T
T0-T(b)
t
f(t)A
T0(a)
.otherwise,
;T,TA
;T,TA
.otherwise,
;T,dtA
;T,dtA
)(f)(fdttf)t(f)(r
T
T
f
0
0
0
0
0
0
2
2
2
0
2
ττ
ττ
τ
τ
ττττ
τ
τ
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Properties of Autocorrelation Functions
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Exercise: Question
PSD
Exercise: Solution
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Crosscorrelation Function• Crosscorrelation function
– A function of τ, not the function of t
–– If s1(t) and s2(t) has the same period T0
– Crosscorrelation and cross PSD are Fourier Transform pairs
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2
2 21
2112
1lim
/T
/T
*
T
*
dt)t(s)t(sT
)t(s)t(s)(R
τ
ττ
)(R)(R ττ 2112
2
2 210
12
0
0
1 /T
/T
* ,dt)t(s)t(sT
)(R τττ
dfe)nff()f(CeC)(R nfj
n
nfj 00 2012
21212
πτπ δτ
Correlation Functions of Power Signals
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)(tf
)(tn
)(fR
)(nR
)(gR
)(tf
)(fgR
Autocorrelation
Cross-correlation
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Properties of Correlation Functions
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• Some properties of correlation functions– Symmetry: and
– Mean-square value:
– Periodicity:
– Average value: and
– Maximum value:
– Additivity:
Orthogonal Uncorrelated
Correlation functions furnish measures of the similarity of a signal either with itself (in the case of autocorrelation) or with another signal (in the case of cross-correlation) versus a relative shift by an amount .
)(tf
τ
)(R)(R gffg ττ
2
2
2 1
0T
TTf dt)t(f)t(f
Tlim)t(f)(R
)(R)T(R)t(f)Tt(f ff ττ
)t(g)t(f)(Rfg τ 2)t(f)(Rf τ
)t(f)(R)(R ff20 τ
)(R)(R)(R)(R)(R)t(y)t(x)t(z yxxyyxz τττττ
)(R)(R ff ττ
0 1 2
2
dtty)t(x
Tlim)(R
T
TTxy ττ 0
2
1
2
1
dtty)t(xdtty)t(xt
t
t
t
Contents
• Deterministic signals
– Classification of signals
– Review of Fourier Transform
– Frequency-domain properties
– Time-domain properties
– Vector space and orthogonality
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Vector Spaces• Set of vectors• Operations on vectors and scalars
– Vector addition: v1 + v2 = v3
– Scalar multiplication: sv1 = v2
– Linear combinations:
• Closed under these operations• Linear independence• Basis• Dimension
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vv
i
n
iia
1
Vector Spaces• Pick a basis, order the vectors in it, then all vectors
in the space can be represented as sequences of coordinates, i.e. coefficients of the basis vectors, in order.
• Example:– Cartesian 3-space（笛卡尔）
– Basis: [i j k]– Linear combination: xi + yj + zk– Coordinate representation: [x y z]– Vector addition:
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]bzazbyaybxax[]zyx[b]zyx[a 212121222111
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Functions as Vectors• Need a set of functions closed under linear
combination, where– Function addition is defined
– Scalar multiplication is defined
• Example:– Quadratic polynomials
– Monomial (power) basis: [x2 x 1]
– Linear combination: ax2 + bx + c
– Coordinate representation: [a b c]
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Metric Spaces• Define a (distance) metric s.t.
– d is nonnegative
– d is symmetric
– Indiscernibles are identical
– The triangle inequality holds
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R)d( 21 v,v
)d()d(: ijjiji v,vv,vVv,v
0 )d(: jiji v,vVv,v
)d()d()d(: kikjjikji v,vv,vv,vVv,v,v
jijiji vvv,vVv,v 0)d(:
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Normed Spaces• Define the length or norm of a vector
– Nonnegative
– Positive definite
– Symmetric
– The triangle inequality holds
• Banach spaces – normed spaces that are complete(no holes or missing points)– Real numbers form a Banach space, but not rational
numbers– Euclidean n-space is Banach
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v
0 vVv :
0vv 0
vvVv aa:Fa,
jijiji vvvvVv,v :
Norms and Metrics• Examples of norms:
– p norm:
• p=1 Manhattan norm
• p=2 Euclidean norm
• Metric from norm• Norm from metric if
– d is homogeneous
– d is translation invariant
then
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ppD
iix
1
1
2121 vvv,v )d(
)d(a)aad(:Fa, jijiji v,vv,vVv,v
)aad()d(:Fa, jijiji v,vv,vVv,v
),d( 0vv
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Inner Product• Define [inner, scalar, dot] product s.t.
• For complex spaces:
• Induces a norm:
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vv,v
Rji v,v
kjkikji v,vv,vv,vv
jiji v,vvv a,a
ijji v,vvv ,
0vv,
0vvv 0,
ijji v,vvv , jiji v,vvv aa,
Examples• Multiplication in R• Dot product in Euclidean N-space
• For real functions over domain [a,b]
• For complex functions over domain [a,b]
• Can add nonnegative weight function
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b
a
dx)x(g)x(fg,f
b
a
dx)x(g)x(fg,f
i
N
ii 2,1,21 vvv,v
1
b
aw
dx)x(w)x(g)x(fg,f
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Hilbert Space• An inner product space that is complete with
respect to the induced norm is called a Hilbert space– Infinite dimensional Euclidean space
– Inner product defines distances and angles
– Subset of Banach spaces
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Orthogonality• Two vectors v1 and v2 are orthogonal if
• v1 and v2 are orthonormal if they are orthogonal and
• Orthonormal set of vectors
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021 v,v
1 2211 v,vv,v
j,iji δv,v (Kronecker delta)
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Example• Linear polynomials over [-1,1]
– B0(x) = 1, B1(x) = x
– Is x2 orthogonal to these?
– Is orthogonal to them?
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01
1
dxx
2
13 2 x
Example• Cosine series
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)ncos()(B),cos()(B,)(B n θθθθθ 10 1
02
1
2
1
02
1
2
0
2
0
2
0
π
π
π
θθ
θθ
θθθ
)])nmsin[()nm(
])nmsin[()nm(
(
nmfor]))nmcos[(])nm(cos[(
d)ncos()mcos(
02
0
2 nmford)n(cos πθθπ
0202
0
2 nmford)(cos πθπ
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Thanks for your kind attention!
Questions?
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