Communication Systems Prapun Suksompong
[email protected] October 19, 2007
Contents 1 Introduction to Signals 1.1 Dirac Delta Function . . . . . . . . . . . . . . . 1.2 Fourier Series . . . . . . . . . . . . . . . . . . . 1.3 Fourier series expansion for real valued function 1.4 Continuous-Time Fourier transform . . . . . . . 1.5 Hilbert Transform . . . . . . . . . . . . . . . . .
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1 1 3 4 6 10
2 Analysis and Transmission of Signals 2.1 Energy Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Power Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 12
3 Modulation
13
4 Sampling Theorem
15
5 Gaussian distribution
17
6 Digital Data Transmission
20
1 1.1
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Introduction to Signals Dirac Delta Function
The (Dirac) delta function or (unit) impulse function is denoted by δ(t). It is usually depicted as a vertical arrow at the origin. Note that δ(t) is not a true function; it is undefined at t = 0. We define δ(t) as a generalized function which satisfies the sampling property (or sifting property) Z φ(t)δ(t)dt = φ(0)
1
for any function φ(t) which is continuous at t = 0. From this definition, It follows that Z (δ ∗ φ)(t) = (φ ∗ δ)(t) = φ(τ )δ(t − τ )dτ = φ(t) where we assume that φ is continuous at t. Intuitively we may δ(t) as a infinitely visualize 1 ε tall, infinitely narrow rectangular pulse of unit area: lim ε 1 |t| ≤ 2 . ε→0
We list some interesting properties of δ(t) here. • δ(t) = 0 when t 6= 0. δ(t − T ) = 0 for t 6= T . R • A δ(t)dt = 1A (0). R (a) δ(t)dt = 1. R (b) {0} δ(t)dt = 1. Rx (c) −∞ δ(t)dt = 1[0,∞) (x). Hence, we may think of δ(t) as the “derivative” of the unit step function U (t) = 1[0,∞) (x). R • φ(t)δ(t)dt = φ(0) for φ continuous at 0. R • φ(t)δ(t − T )dt = φ(T ) for φ continuous at T . In fact, for any ε > 0, Z
T +ε
φ(t)δ(t − T )dt = φ(T ). T −ε
• δ(at) =
1 δ(t) |a|
• δ(t − t1 ) ∗ δ(t − t2 ) = δ (t − (t1 + t2 )). • Fourier properties: ◦ Fourier series: δ(x − a) = ◦ Fourier transform: δ(t) =
1 2π
R
+
1 π
∞ P
cos(n(x − a)) on [−π, π].
k=1 j2πf t
e
df
• For a function g whose real-values roots are ti , δ (g (t)) =
n X δ (t − ti ) k=1
|g 0 (ti )|
(1)
[2, p 387]. Hence, Z f (t)δ(g(t))dt =
X x:g(x)=0
2
f (x) . |g 0 (x)|
(2)
Note that the (Dirac) delta function is to be distinguished from the discrete time Kronecker delta function. As a finite measure, δ is a unit R massRat 0; that is for any set A, we have δ(A) = 1[0 ∈ A]. In which case, we have again gdδ = f (x)δ(dx) = g(0) for any measurable g. For a function g : D → Rn where D ⊂ Rn , X
δ(g(x)) =
z:g(z)=0
δ(x − z) |det dg(z)|
(3)
[2, p 387].
1.2
Fourier Series
Let the (real or complex) signal r (t) be a periodic signal with period T0 . Suppose the following Dirichlet conditions are satisfied (a) r (t) is absolutely integrable over its period; i.e.,
RT0
|r (t)|dt < ∞.
0
(b) The number of maxima and minima of r (t) in each period is finite. (c) The number of discontinuities of r (t) in each period is finite. ∞
Then r (t) can be expanded in terms of the complex exponential signals (ejnω0 t )n=−∞ as r˜ (t) =
∞ X
jnω0 t
cn e
= c0 +
n=−∞
∞ X
ck ejkω0 t + c−k e−jkω0 t
(4)
k=1
where ω0 = 2πf0 =
2π , T0
α+T0
1 ck = T0
Z
r (t) e−jkω0 t dt,
α
for some arbitrary α. In which case, ( r (t) , r˜ (t) = r(t+ )+r(t− ) 2
if r (t) is continuous at t , if r (t) is not continuous at t
We give some remarks here. • The parameter α in the limits of the integration is arbitrary.It Rcan be chosen to simplify computation of the integral. We can simply write ck = T10 r (t) e−jkω0 t dt to T0
emphasize that we only need to integrate over one period of the signal; the starting point is not important. 3
• The coefficients ck =
1 T0
r (t) e−jkω0 t dt are called the (k th ) Fourier (series) coef-
R T0
ficients of (the signal) r (t). These are, in general, complex numbers. R • c0 = T10 r (t) dt = average or DC value of r(t) T0
• The Dirichlet conditions are only sufficient conditions for the existence of the Fourier series expansion. For some signals that do not satisfy these conditions, we can still find the Fourier series expansion. • The quantity f0 = T10 is called the fundamental frequency of the signal r (t). The nth multiple of the fundamental frequency (for positive n’s) is called the nth harmonic. • ck ejkω0 t + c−k e−jkω0 t = the k th harmonic component of r (t). k = 1 ⇒ fundamental component of r (t). 1 T0
1.1. Parseval’s Identity: Pr =
R
∞ P
|r (t)|2 dt =
|ck |2
k=−∞
T0
1.2. Real, Odd/Even properties • If r(t) is even (r(−t) = r(t)), then c−k = ck . • If r(t) is odd (r(−t) = −r(t)), then c−k = −ck . • If r(t) is real valued and even, then so is ck as a function of k. • If r(t)is real-valued and odd, then ck ’s are pure imaginary and c−k = −ck
1.3
Fourier series expansion for real valued function
Suppose r (t) in the previous section is real-valued; that is r∗ = r. Then, we have c−k = c∗k and we provide here three alternative ways to represent the Fourier series expansion: r˜ (t) =
∞ X
jnω0 t
cn e
= c0 +
n=−∞
= c0 +
∞ X
ck ejkω0 t + c−k e−jkω0 t
(5)
k=1
∞ X
(ak cos (kω0 t)) +
k=1 ∞ X
= c0 + 2
∞ X
(bk sin (kω0 t))
(6)
k=1
|ck | cos (kω0 t + ∠ck )
k=1
4
(7)
where the corresponding coefficients are obtained from α+T0
1 ck = T0
Z
r (t) e−jkω0 t dt =
1 (ak − jbk ) 2
(8)
α
2 ak = 2Re {ck } = T0
Z r (t) cos (kω0 t) dt
(9)
T
0 Z 2 r (t) sin (kω0 t) dt bk = −2Im {ck } = T0 T0 q |ck | = a2k + b2k bk ∠ck = − arctan ak a0 c0 = 2
(10) (11) (12) (13) (14)
The Parseval’s identity can then be expressed as Z ∞ ∞ X X 1 2 2 2 |ck |2 |ck | = c0 + 2 Pr = |r (t)| dt = T0 k=1 k=−∞ T0
1.3. Examples: • Train of impulses: ∞ X
∞ ∞ 1 X jkω0 t 1 2 X δ (t − kT0 ) = δTs (t) = e = + cos kω0 t T T T 0 0 0 k=−∞ k=−∞ k=1
T0
2T0
(15)
t
Figure 1: Train of impulses • Square pulse periodic signal: 1 2 1 1 1 cos ω0 t − cos 3ω0 t + cos 5ω0 t − cos 7ω0 t + . . . 1 [cos ω0 t ≥ 0] = + 2 π 3 5 7 • Bipolar square pulse periodic signal: 4 1 1 1 sgn(cos ω0 t) = cos ω0 t − cos 3ω0 t + cos 5ω0 t − cos 7ω0 t + . . . π 3 5 7 5
-1
1
−T0
t
T0
Figure 2: Square pulse periodic signal 1
−T0
t
T0 -1
Figure 3: Bipolar square pulse periodic signal
1.4
1 Continuous-Time Fourier transform
The (direct) Fourier transform of g(t) is defined by −T0
ˆ (ω) = G
t
T0
Z∞
g (t) e−jωt dt.
(16)
−∞
We may simply write G = F {g}. Sometimes the magnitude and phase of G are shown explicitly by writing G = |G| ejθg where both |G| and θg are real-valued functions of ω. 1 2π
Z∞
F
ˆ (ω) ejωt dω = g (t) ) ˆ (ω) = −− * G − −G −1
Z∞
F
−∞
g (t) e−jωt dt
−∞
ˆ are given by fourier and ifourier. Note also that G(0) = RIn MATLAB, these identities R 1 g(t)dt and g(0) = 2π G(ω)dt. 1.4. Conjugate and Time Inversion (Time Reversal): F ˆ (−ω) * g (−t) − )− − −G −1 F F
ˆ ∗ (−ω) * g (t) − )− − −G −1 ∗
F F
ˆ ∗ (ω) * g ∗ (−t) − )− − −G −1 F
1.5. Shifting properties F
−− * • Time-shift: g (t − t1 ) ) − − e−jωt1 G (ω) −1 F
F ˆ (ω − ω1 ) * • Frequency-shift (or modulation): ejω1 t g (t) − )− − −G −1 F
6
1.6. Unit impulse: F
* ejω0 t − )− − − 2πδ (ω − ω0 ) −1
(17)
F
∞ X
jkω0 t
ck e
F
k=−∞
∞ X
F
− * )− − − −1
2πck δ (ω − kω0 )
(18)
k=−∞
F
* δ (t − t0 ) − )− − − e−jωt0 −1
(19)
* δ (t) − )− − −1 −1
(20)
F F F F
−− * 1) − − 2πδ (ω) −1
(21)
−− * a) − − a2πδ (ω) −1
(22)
F F F
Property (18) is of importance because it shows transform of periodic signal which is expressed in its Fourier series form (as in (4)). A special case is when the signal is a train of impulses: ∞ X
∞ X
F
−− * δ(t − nT0 ) ) − − ω0 −1 F
n=−∞
δ(ω − nω0 ) where ω0 =
n=−∞
2π . T0
F ˆ 1 (ω) + G ˆ 2 (ω). −− * 1.7. Linearity: c1 g1 (t) + c2 g2 (t) ) − − c1 G −1 F
F
* 1.8. Time-scaling rule: g (at) − )− − − −1 F
F
* 1.9. Re {g (t)} − )− − − −1 F
1 2
1 ˆ G |a|
ω a
ˆ (ω) + G ˆ ∗ (−ω) G
1.10. Convolution F ˆ 1 (ω) · G ˆ 2 (ω) * • Convolution-in-time Rule: g1 (t) ∗ g2 (t) − )− − −G −1 F
F
1 ˆ G 2π 1
−− * • Convolution-in-frequency Rule: g1 (t) · g2 (t) ) − − −1 F
F
F
ˆ 2 (ω) (ω) ∗ G
−− * −− * 1.11. Duality: Suppose f (t) ) − − 2πf (−ω). − − g (ω). Then, g (t) ) −1 −1 F
1.12. Parseval’s theorem:
R∞ −∞
|g (t)|2 dt =
F
1 2π
2 R∞ ˆ G (ω) dω
−∞
7
1.13. Unit step: F 1 −− * u (t) = 1 [t ≥ 0] ) + πδ (ω) − − F −1 jω F 2 1, t > 0 −− * sgn t = )− − −1, t < 0 F −1 jω j F −− ) −* − sgn(ω) πt F −1 Z t F 1 * (g ∗ u)(t) = g(τ )dτ − G(ω) + πG(0)δ(ω) )− − − −1 F jω −∞
(23) (24) (25) (26)
So, if g (or equivalently, G) is band-limited to |ω| ≤ B, then g ∗ u is also bandlimited to |ω| ≤ B. 1.14. Exponential: Assume α, σ > 0. 1 F α + jω F 1 * eαt u (−t) − )− − − −1 F α − jω F 2α −− * e−α|t| ) − − F −1 α2 + ω 2 F 1 * te−αt u (t) − )− − − −1 F (α + jω)2 F n! * tn e−αt u (t) − )− − − −1 F (α + jω)n+1 r F 1 π 2 2 * ke−αt − e−( 4α )ω )− − − k −1 F α F
−− * e−αt u (t) ) − − −1
1.15. Modulation: F
* cos (ωc t + θ) − )− − − πδ (ω − ωc ) ejθ + πδ (ω + ωc ) e−jθ −1 F F
π (δ (ω − ω0 ) − δ (ω + ω0 )) F j F 1ˆ 1ˆ * gωc ,θ (t) = g (t) × cos (ωc t + θ) − (ω − ωc ) ejθ + G (ω + ωc ) e−jθ )− − − G −1 F 2 2 F 1 ˆ 1 ˆ * g (t) × sin (ωc t + θ) − (ω + ωc ) e−jθ )− − − G (ω − ωc ) ejθ − G −1 F 2j 2j Suppose g is bandlimited; that is G = 0 for |ω| > ωg = 2πBg . If ωc > ωg , then the ˆ (ω − ωc ) and G ˆ (ω + ωc ) are disjoint, and hence they are orthogonal in support sets of G the frequency domain and their energy added. In which case, Egωc ,θ = 21 Eg . −− * sin (ω0 t) ) − − −1
1.16. Rectangular and Sinc: Assume a, ω0 > 0. F 2 sin (aω) −− * = 2a sinc (aω) 1 [|t| ≤ a] ) − − F −1 ω ω0 sin (ω0 t) F − sinc (ω0 t) = )− −* − 1 [|ω| ≤ ω0 ] F −1 π πt 8
sinc ( 2π f 0t )
⎛T ⎞ T0sinc ⎜ 0 ω ⎟ ⎝2 ⎠ T0
1 1 ⎡ ω ≤ 2π f 0 ⎤⎦ 2 f0 ⎣ 1 2 f0
1
1 2 f0
2π T0
ω 1
2π f 0
F t YZZZ ZZZX F −1
−
1 2 f0
1 f0
T0 2
T0 2
t
F ZZZX YZZZ F −1
T0sinc (π T0 f
ω
)
T0
T0
1 T0
f
f
f0
Figure 4: Fourier transform of sinc and rectangular functions 1.17. Derivative rules: • Time-derivative rule:
d g dt
F ˆ (ω) * (t) − )− − − jω G −1 F
F
−− * • Frequency-derivative rule: −jtg (t) ) − − −1 F
d ˆ G (ω) dω
1.18. Real, Odd, and Even ˆ (−ω) = • Conjugate Symmetry Property: If g (t) is real-valued (g (t) = g ∗ (t)), then G ∗ ˆ (ω). In particular, |G| is even and θg is odd. G ˆ (ω) is also even (G ˆ (−ω) = G ˆ (ω)) • If g (t) is even (g (t) = g (−t)), then G ˆ (ω) is also odd (G ˆ (−ω) = −G ˆ (ω)) • If g (t) is odd (g (t) = −g (−t)), then G ˆ (ω). • If g (t) is real and even, then so is G ˆ (ω) is pure imaginary and odd. • If g (t) is real and odd, then G 1.19. A signal cannot be simultaneously time-limited and band-limited. Proof. Suppose g(t) is simultaneously (1) time-limited to T0 and (2) band-limited to B. Pick any positive number Ts and positive integer K such that fs = T1s > 2B and K > TT0s . The sampled signal gTs (t) is given by gTs (t) =
X
g[k]δ (t − kTs ) =
k
K X
g[k]δ (t − kTs )
k=−K
where g[k] = g (kTs ). Now, because we sample the signal faster than the Nyquist rate, we can reconstruct the signal g by producing gTs ∗ hr where the LPF hr is given by Hr (ω) = Ts 1[ω < 2πfc ] with the restriction that B < fc <
1 Ts
G(ω) =
− B. In frequency domain, we have K X
g[k]e−jkωTs Hr (ω).
k=−K
9
Consider ω inside the interval I = (2πB, 2πfc ). Then, 0
ω>2πB
=
G(ω)
ω<2πfc
=
Ts
K X
g (kTs ) e−jkωTs
z=ejωTs
=
k=−K
Ts
K X
g (kTs ) z −k
(27)
k=−K
Because z 6= 0, we can divide (27) by z −K and then the last term becomes a polynomial of the form a2K z 2K + a2K−1 z 2K−1 + · · · + a1 z + a0 . By fundamental theorem of algebra, this polynomial has only finitely many roots– that is there are only finitely many values of z = ejωTs which satisfies (27). Because there are uncountably many values of ω in the interval I and hence uncountably many values of z = ejωTs which satisfy (27), we have a contradiction. ˆ (Ω) to distinguish it from 1.20. Sometimes, the Fourier transform above is denoted by G the DTFT. In which case, Z∞
1 2π
Z∞
F
ˆ (Ω) ejΩt dΩ = g (t) − ˆ (Ω) = * G )− − −G −1 F
−∞
g (t) e−jΩt dt.
−∞
Some references define
Z∞ G (f ) =
g (t) e−j2πf t dt.
(28)
−∞
In which case, we have Z∞
j2πf t
G (f ) e
F
* df = g (t) − )− − − G (f ) = −1
Z∞
F
g (t) e−j2πf t dt.
−∞
−∞
This definition eliminates several extra π and 2π factors in the identities resulting from the definition (16). Of course, (16) and (28) are related by ˆ ˆ (Ω) = G (f )| Ω . G (f ) = G (Ω) and G f= 2π
Ω=2πf
1.5
Hilbert Transform
The Hilbert transform of a real-valued function g, denoted by H[g] is given by g ∗ h where π 1 h(t) = πt . Hence, H(ω) = −j sgn(ω) = e−j 2 sgn(ω) and Z F 1 g(τ ) H {g} (t) = dτ − )− −* − F {H {g}} (t) = −jG(ω) sgn(ω). F −1 π t−τ Note that • h is non-causal and hence unrealizable; 10
• H[g] is still a real-valued function. 1.21. A Hilbert transformer H is an ideal phase shifter that shifts the phase of every spectral component by − π2 . For example, it transform cos to sin in (29). 1.22. Because H 2 = −1, we have H [H[g]] = −g and thus the inverse Hilbert transform of a real-valued function gˆ is given by gˆ ∗ (−h). 1.23. Examples H
cos(ω0 t + φ) − → sin(ω0 t + φ)
(29)
H
sin(ω0 t + φ) − → − cos(ω0 t + φ) H 1 − cos(t) sinc(t) − → t 1 H 1 t + 12 1 |t| ≤ − → ln 2 π t− 1
(30) (31) (32)
2
2
Analysis and Transmission of Signals
For a signal g(t), the instantaneous power p(t) dissipated in the 1-Ω resister is pg (t) = |g(t)|2 regardless of whether g(t) represents a voltage or a current. To emphasize the fact that this power is based upon unity resistance, it is often referred to as the normalized power. The total energy of the signal g(t) is then Z Eg = |g(t)|2 dt and the average power is given by 1 Pg = lim T →∞ T
ZT /2
1 |g (t)| dt = lim T →∞ 2T 2
Z
−T /2
T
|g(t)|2 dt.
−T
If Eg is finite and nonzero, g is referred to as an energy signal. If pg is finite and nonzero, g is referred to as a power signal. Note that the power signal has infinite energy and an energy signal has zero average power; thus the two categories are mutually exclusive.
2.1
Energy Signal
2.1. Definitions • Energy: Eg =
R
|g(t)|2 dt.
• Energy spectral density (ESD): Ψg (t) = |G(ω)|2 . ◦ ESD is a positive, real, and even function of ω. 11
• Time autocorrelation function: Z ψg (τ ) = g ∗ (µ)g(µ + τ )dµ = g ∗ (τ ) ∗ g(−τ ) Z = g(µ)g ∗ (µ − τ )dµ = g(τ ) ∗ g ∗ (−τ ) ◦ ψg is invariant to time-shift in g: Suppose h(t) = g(t − t0 ), then ψg = ψh . 2.2. Z
|g(t)|2 dt Z Z Z 1 1 2 = |G(ω)| dω = Ψg (ω)dω = Ψg (2πf )df 2π 2π
Eg =
F
2.3. ψg (τ ) − → Ψg (ω) 2.4. Example • For g(t) = 1[t0 ,t0 +T ] (t), we have ψg (τ ) = 1 −
|τ | T
1[−T,T ] (τ ) and Ψg (ω) = T sinc2
ωT 2
.
2.5. Suppose g and y are the input and output signals of an LTI system with transfer function H g → LTI : H → y , then Ψy (ω) = |H(ω)|2 Ψg (ω).
2.2
Power Signal
2.6. Definitions: • gT (t) = g (t) 1 |t| ≤ T2 1 T →∞ T
• Power: Pg = lim
TR/2
|g (t)|2 dt = lim
1 EgT T →∞ T
−T /2
1 T →∞ T
• Power spectral density (PSD): Sg (ω) = lim
= hg 2 i. |GT (ω)|2 = lim
1 2 ΨgT (t) T →∞ T
◦ PSD represents the power per unit bandwidth (in Hz) of the spectral components at the frequency ω. ◦ PSD is a positive, real, and even function of ω. • Time autocorrelation function: 1 Rg (τ ) = lim T →∞ T
ZT /2
1 g (µ) g (µ + τ )dµ = lim T →∞ T ∗
−T /2
ZT /2 −T /2
∗
∗
= hg (·) g (· + τ )i = hg (·) g (· − τ )i ◦ Rg (−τ ) = Rg∗ (τ ) 12
g (µ) g ∗ (µ − τ )dµ
2.7. ZT /2
1 1 Pg = lim |g (t)|2 dt = lim EgT = g 2 T →∞ T T →∞ T −T /2 Z Z 1 = Sg (ω)dω = Sg (2πf )df 2π F
2.8. Rg (τ ) − → Sg (ω) R T /2 2.9. Rg1 g2 (τ ) = lim −T /2 g1 (t)g2 (t + τ )dt T →∞
• If g = g1 + g2 , then Rg = Rg1 + Rg2 + Rg1 g2 + Rg2 g1 . 2.10. Examples • g(t) = a cos (ω0 t + θ) ◦ Rg (τ ) = 12 a2 cos ω0 t ◦ Sg (ω) = π2 a2 (δ(ω − ω0 ) + δ(ω + ω0 )) P • Periodic function r(t) = d0 + ∞ n=1 dn cos (nω0 t + θn ) P 2 ◦ Rr (τ ) = d20 + 21 ∞ n=1 dn cos nω0 τ P 2 ◦ Sr (ω) = 2πd20 δ(ω) + π2 ∞ n=1 dn (δ(ω − nω0 ) + δ(ω + nω0 )) 2.11. Suppose g and y are theinput and output signals of an LTI system with transfer function H g → LTI : H → y , then Sy (ω) = |H(ω)|2 Sg (ω).
3
Modulation
Let the carrier frequency be at fc [Hz] with corresponding angular frequency ωc = 2πfc . 3.1. Recall from (1.15) that F 1ˆ 1ˆ jθ −jθ −− * gωc ,θ (t) = g (t) × cos (ωc t + θ) ) G (ω − ω . − − c ) e + G (ω + ωc ) e −1 F 2 2
Furthermore, if (1) g is bandlimited to |ω| ≤ ωg = 2πBg and (2) |ω| > ωg = 2πBg , then Egωc ,θ = 21 Eg . This is not true for non-bandlimited g. For example, take g = 1[0,T ] , then Z Eg 1 g 2 (t) cos2 (ωc t)dt = + cos (ωc T ) sin (ωc T ) 2 2ω where the second term does not vanish for all ωc . It will vanish when ωc → ∞.
13
3.2. To produce the modulated signal g(t) cos ωc t, we can (1) multiply g(t) by “any” periodic and even signal r(t) whose period is Tc = 2π and (2) use bandpass filter to restrict ωc frequency contents to around ωc . In particular, because r(t) is an even function, we know that ∞ X r (t) = c0 + ak cos (kωc t). k=1
Therefore, m(t)r (t) = c0 m(t) +
∞ X
ak m(t) cos (kωc t).
k=1
m (t )
M (ω ) A
2π B
×
m ( t ) cos (ωct )
BPF
r (t )
F {m × r}(ω ) 1 Aa c0 A 2 1 −2ωc
−ωc
ωc 2π B
1 Aa2 2 2ωc
ωc − 2π B
BPF
Figure 5: Modulation of m(t) via even and periodic r(t) In general, for this scheme to work, we need • a1 6= 0; that is Tc is the “least” period of r; • ωc > 4πB; that is fc > 2B (to prevent overlapping). 3.3. Double-sideband suppressed carrier (DSB-SC) modulation: LPF {(m (t) cos ωc t) 2 cos ((ωc + ∆ω) t + θ)} = m (t) cos ((∆ω) t + θ) • Need ωc ≥ 2πB • The modulated signal spectrum centered at ωc is composed of two parts: a portion that lies above ωc , known as the upper sideband (USB), and a portion that lies below ωc , known as the lower sideband (LSB). Hence, this is a modulation scheme with double sidebands. • The modulated signal does not contain a discrete component of the carrier frequency ωc . 3.4. Amplitude Modulation (AM): ϕAM (t) = (A + m (t)) cos ωc t = A cos ωc t + m (t) cos ωc t | {z } | {z } carrier
14
sidebands
Modulator m (t )
Demodulator m ( t ) cos ωct
⊗ (Modulated signal)
(Modulating signal)
⊗
cos ωct (Carrier)
1 1 M (ω ) + ( M (ω + 2ωc ) + M (ω − 2ωc ) ) 2 4 A2 LPF
−ωc
LSB
USB
LSB
USB
A2
2π B
mˆ ( t )
cos ωct (Carrier) 1 ( M (ω + ωc ) + M (ω − ωc ) ) 2
M (ω ) A
LPF
ω
ωc
−ωc
ωc 2ωc
ω
4π B
Figure 6: DSB-SC modulation and demodulation 3.5. QAM: ϕQAM (t) = m1 (t) cos (ωc t) + m2 (t) sin (ωc t) LPF {ϕQAM (t) 2 cos ((ωc + ∆ω) t + δ)} = m1 (t) cos ((∆ω) t + δ) − m2 (t) sin ((∆ω) t + δ) LPF {ϕQAM (t) 2 sin ((ωc + ∆ω) t + δ)} = m1 (t) sin ((∆ω) t + δ) + m2 (t) cos ((∆ω) t + δ)
4
Sampling Theorem
A low-pass signal g whose spectrum is band-limited to B Hz (G(ω) = 0 for |ω| > 2πB) can be reconstructed exactly (without any error) from its sample taken uniformly at a rate (sampling frequency) Rs > 2B Hz (samples per second). 4.1. The “sampling” can be done by producing gTs (t) = g (t) rTs (t) where rTs (a) is periodic with period Ts =
1 Rs
<
1 2B
(b) has nonzero mean. 4.2. Signal Reconstruction: Because rTs is periodic, it has fourier series expansion ∞ X
r˜ (t) =
cn ejnω0 t
n=−∞
where ωs = 2πfs =
2π . Ts
Hence, GTs (ω) =
X
cn G (ω − nωs ).
n
15
g (t )
× rTs ( t )
G (ω ) A
gˆ ( t )
LPF
{
F g × rTs
} (ω )
c0 A
2π B
ω
−2ωs
−ωs
ωs 2π B
LPF
2ωs
ω
ωs − 2π B
Figure 7: Sampling and Reconstruction Suppose 2πB < ωs − 2πB (or equivalently Rs > 2B), then there is no overlapping and we can get G back by LPF H with cutoff fc ∈ [B, Rs − B). More specifically, F −1 ωc 1 2fc 1 [|ω| ≤ ωc ] − sinc (ωc t) = sinc (2πfc t) . )− −* − h (t) = F c0 c0 π c0
H (ω) =
4.3. Interpolation formula: Suppose rTs is a train of impulses δTs as in (15). In which case, X g [k] δ (t − kTs ) gTs (t) = k
where g [k] = g (kTs ). Note that we have c0 =
1 Ts
= fs . Therefore,
F −1
H (ω) = Ts 1 [|ω| ≤ ωc ] − )− −* − h (t) = F
2fc sinc (2πfc t) . fs
The filtered output gˆ = gTs ∗ h which is g can now be expressed as a sum g (t) =
X
g [k] h (t − kTs ) =
k
2fc X g [k] sinc (2πfc (t − kTs )) fs k
Furthermore, suppose we choose fs = 2B and fc = B, then we have H (ω) =
F −1 1 −− 1 [|ω| ≤ 2πB] ) −* − h (t) = sinc (2πBt) . F 2B
In which case, g (t) =
X
g [k] sinc (2πB (t − kTs )) =
k
X
g [k] sinc (2πBt − kπ).
k
4.4. A band pass signal whose spectrum exists over a frequency band fc − B2 < |f | < fc + B2 ha s a bandwidth B Hz. Such a signal is uniquely determined by 2B samples per second. The sampling scheme uses two interlaced sampling trains, each at a rate of B samples per second (known as second-order sampling). 16
5
Gaussian distribution
5.1. Gaussian distribution: (a) Denoted by N (m, σ 2 ) . N (0, 1) is the standard Gaussian (normal) distribution. 2
(b) fX (x) =
1 x−m √ 1 e− 2 ( σ ) 2πσ
.
(c) FX (x) = normcdf(x,m,sigma). • The standard normal cdf is sometimes denoted by Φ(x). It inherits all properties of cdf. Moreover, note that Φ(−x) = 1 − Φ(x). 1 jvX 1 2∞ 2 − jω m − ω σ v σ 2 (d) ϕX23) (v) = E transform: e =F ejmv− Fourier . ( f X )2= ∫ f.]X ( x ) e− jω x dt = e 2
2
−∞
1 2 2
(e) MX (s) = esm+∞2 s−ασx 24) Note that
∫e
2
π . α
dx =
−∞
R∞
(f) Fourier transform: F {f } = ⎛ x−m⎞ X 25) P [ X > x ] = Q ⎜ ⎝ σ
1
−jωm− 2 ω fX (x)⎛ ex−jωx − m ⎞dt =⎛ e x − m ⎞
x] = 1 − Q ⎜ ⎟ ; P [ X <−∞ ⎠ ⎝ σ
⎟ = Q⎜− ⎠ ⎝ σ
2 σ2
.
⎟. ⎠
x−m (g) P [X •> x] ≥⎤ x] = Q Px−m = 1> σ−⎤Φ = Φ − x−m P ⎡⎣=X P − μ[X< σ = 0.6827, ⎡ X − μ = 0.3173 σ σ σ ⎦ ⎣ ⎦ x−m x−m P [X < x] = P [X ≤ x] = 1 − Q x−m = Q − = Φ . σ σ σ P ⎡⎣ X − μ > 2σ ⎤⎦ = 0.0455, P ⎡⎣ X − μ < 2σ ⎤⎦ = 0.9545 fX ( x)
fX ( x)
95%
68%
μ −σ
μ μ +σ
μ − 2σ
μ
μ + 2σ
∞ 1 −x : Q (8: corresponds to P [ X of 26) Q-function z ) =Probability > zX ~ Nσ 2( 0,1 ] where Figure function ∼ N X(m, ) . ); ∫z 2π e 2 dx density that is Q ( z ) is the probability of the “tail” of N ( 0,1) . 2
5.2. Properties
N ( 0,1)
(a) P [|X − µ| < σ] = 0.6827; P [|X − µ| > σ] = 0.3173 P [|X − µ| > 2σ] = 0.0455; P [|X − µ| < 2σ] = 0.9545 (b) Moments and central moments: h
k
(i) E (X − µ) h
i
h
k−2
= (k − 1) E (X − µ) 0
i
Q( z)
z
i
=
0.5 0, k odd k 1 · 3 · 5 · · · · · (k − 1) σ , k even
(
q 1 1) σ k 2 , k odd 2 · 4 · 6 · · · · · (k − π = function with Q ( 0 ) = 2 . k 1 · 3 · 5 · · · · · (k − 1) σ , k even
k decreasing
a) |X Q is− a µ| (ii) E
b) Q ( − z ) = 1 − Q ( z )
2 (iii) Var [X ] = 4µ2 σ 2 + 2σ 4 . −1
c) Q
(1 − Q ( z ) ) = − z π
d) Q ( x ) =
1
2
e π∫ 0
π −
x2 2 sin θ 2
1
4
−
x2
2 sin θ dθ . Q ( x ) = ∫ e 17 dθ . 2
π
2
2
0
d 1 − x2 d 1 − Q ( x) = − e ; Q ( f ( x)) = − e e) dx dx 2π 2π
( f ( x )) 2
2
d f ( x) . dx
n EX n E [(X − µ)n ]
0 1 1
1 µ 0
2 µ2 + σ 2 σ2
3 µ (µ2 + 3σ 2 ) 0
4 µ4 + 6µ2 σ 2 + 3σ 4 3σ 4
(c) For N (0, 1) and k ≥ 1, E X k = (k − 1) E X k−2 =
0, k odd 1 · 3 · 5 · · · · · (k − 1) , k even.
The first equality comes from integration by parts. Observe also that (2m)! E X 2m = m . 2 m! (d) L´evy–Cram´er theorem: If the sum of two independent non-constant random variables is normally distributed, then each of the summands is normally distributed. • Note that
R∞
2
e−αx dx =
pπ α
−∞
.
5.3 (Length bound). For X ∼ N (0, 1) and any (Borel) set B, Z
|B|/2
P [X ∈ B] ≤
fX (x) = 1 − 2Q
−|B|/2
|B| 2
,
where |B| is the length (Lebesgue measure) of the set B. This is because the probability is concentrated around 0. More generally, for X ∼ N (m, σ 2 ) |B| P [X ∈ B] ≤ 1 − 2Q . 2σ 5.4 (Stein’s Lemma). Let X ∼ N (µ, σ 2 ), and let g be a differentiable function satisfying E |g 0 (X)| < ∞. Then E [g(X)(X − µ)] = σ 2 E [g 0 (X)] . [1, Lemma 3.6.5 p 124]. Note that this is simply integration by parts with u = g(x) and dv = (x − µ)fX (x)dx. • E (X − µ)k = E (X − µ)k−1 (X − µ) = σ 2 (k − 1)E (X − µ)k−2 . 5.5. Q-function: Q (z) =
R∞ z
2
x √1 e− 2 2π
dx corresponds to P [X > z] where X ∼ N (0, 1);
that is Q (z) is the probability of the “tail” of N (0, 1). The Q function is then a complementary cdf (ccdf). (a) Q is a decreasing function with Q (0) = 21 . (b) Q (−z) = 1 − Q (z) = Φ(z) 18
N ( 0,1)
1 0.9 0.8 0.7
Q(z)
0.6 0.5 0.4 0.3 0.2 0.1
0
0 -3
z
-2
-1
0
1
2
3
z
Figure 9: Q-function (c) Q−1 (1 − Q (z)) = −z π
(d) Craig’s formula: Q (x) =
1 π
R2
π
−
e
x2 2 sin2 θ
dθ =
0
1 π
R2
x2
e− 2 cos2 θ dθ, x ≥ 0.
0
i.i.d.
To see this, consider X, Y ∼ N (0, 1). Then, π
Z2 Z∞
ZZ Q (z) =
fX,Y (r cos θ, r sin θ)drdθ.
fX,Y (x, y)dxdy = 2 0
(x,y)∈(z,∞)×R
z cos θ
where we evaluate the double integral using polar coordinates [3, Q7.22 p 322]. π
2
(e) Q (x) = (f)
d Q (x) dx
(g)
d Q (f dx
(h)
R
1 π
R4
x2
e− 2 sin2 θ dθ
0 x2
= − √12π e− 2
(x)) = − √12π e−
(f (x))2 2
d f dx
(x)
R
Q (f (x)) g (x)dx = Q (f (x)) g (x)dx +
R
(f (x)) √1 e− 2 2π
(i) P [X > x] = Q x−m σ x−m P [X < x] = 1 − Q x−m = Q − . σ σ (j) Approximation: h i z2 1 √ (i) Q (z) ≈ (1−a)z+a z2 +b √12π e− 2 ; (ii) 1 −
1 x2
e− x22
(iii) Q (z) ≈
√
x 2π √1 z 2π
a = π1 , b = 2π
x2
≤ Q (x) ≤ 12 e− 2 − z2 1 − 0.7 e 2 ;z > 2 2 z 19
2
d f dx
Rx (x) g (t) dt dx a
27) Moment and central moment n
0
1
2
3
4
2 2 + 3σ 4 1 μ μ 2 + σ 2 μ ( μ z+ 3σ ) μ 4 + 6μ 2σ 2 √ EX n R 2 5.6. Error function (MATLAB): erf (z) = √2π e−x dx = 1 − 2Q 2z n 00 E ⎡( X − μ ) ⎤ 1 0 σ2 3σ 4 ⎣ ⎦ (a) It is an odd function of z. k odd ⎧0, k k −2 k • E ⎡( X − μ ) ⎤ = ( k − 1) E ⎡( X − μ ) ⎤ = ⎨ 1 ⎣ ⎦ ⎣ ⎦ (b) For z ≥ 0, it corresponds to P [|X| < z] where ⋅5∼ ⋅ N ⋅ ( k 0, − 12)σ. , k even ⎩1 ⋅ 3X
⎧ (c) lim erf (z) = 1 2 k z→∞ , k odd k ⎪ 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ( k − 1) σ ⎡ ⎤ • E X −μ =⎨ [Papoulis p 111]. π ⎦ (d) erf (−z)⎣= −erf (z) ⎪1 ⋅ 3 ⋅ 5 ⋅ ⋅ ( k − 1) σ k , k even ⎩ 1 z 2 21 √z (e) Q (z) ⎡⎣ X 2 ⎤⎦ √ Var2 erfc σ 2 + 12σ−4 erf =24 μ= • = . 2
k −2 k odd ⎧0, x , E ⎡ X k ⎤1= ( k − 1) √ 1 ( 0,1) and k √ 28) For ≥ 1 N = ⎦2 erfc − Ex2⎡⎣ X ⎤⎦ = ⎨1 ⋅ 3 ⋅ 5 ⋅ ⋅ k − 1 , k even (f) Φ(x) = 2 1 + erf ⎣ (2) ( ) ⎩ √ z −1 (2q) erf ( z ) = 2 e− x2 dx = 1 − 2Q 2 z corresponds to (g)29) Q−1 2 erfc(Matlab): (q) = Error function π ∫0 √ R∞ 2 (h) The complementary error function: erfc (z) = 1−erf (z) = 2Q 2z = √2π z e−x dx ⎛ 1⎞ P ⎡⎣ X < z ⎤⎦ where X ~ N ⎜ 0, ⎟ . ⎝ 2⎠
(
)
⎛ 1⎞ N ⎜ 0, ⎟ ⎝ 2⎠
erf ( z ) Q
(
2z
)
z
0
a) lim erf ( z ) = 1 Figure 10: erf-function and Q-function z →∞ b) erf ( − z ) = −erf ( z )
6
Digital Data Transmission
The pulses p(t) are transmitted at a rate of Rb =
1 Tb
pulses per second.
6.1. Line coding or transmission coding Transmit “1” by
Transmit “0” by
Polar
p(t)
−p(t)
On-off
p(t)
no pulse
Bipolar
alternate ±p(t)
no pulse
• Assume “1” and “0” are equally likely. 20
P(E) Q Aσnp Ap Q 2σ n Ap 3 Q 2σ 2 n
Received power αA2p + σn2 α 2 A 2 p
+ σn2
α 2 A 2 p
+ σn2
• Bipolar signaling is also known as pseudoternary or alternate mark. • Received power is calculated by assuming that p(t) is rectangular whose width is αTb where α ∈ (0, 1]. ◦ return-to-zero (RZ): α ∈ (0, 1) ◦ nonreturn-to-zero (NRZ): α = 1 • The polar scheme is the most power efficient code; for a given noise immunity (error probability), this code requires the least power. See also figure reflineCoding. 0
10
Polar On-off Bipolar
-2
Probability of error
10
-4
10
Bipolar -6
10
Polar
On-off
-8
10
-10
10
0
10
20
30
40
50
Normalized Received power
60
70
80
⎞ 1 ⎛ PR ⎜ 2 − 1⎟ α ⎝ σn ⎠
Figure 11: Line codes: Power vs. Error Probability
References [1] George Casella and Roger L. Berger. Statistical Inference. Duxbury Press, 2001. 5.4 [2] Donald G. Childers. Probability And Random Processes Using MATLAB. McGraw-Hill, 1997. 1.1, 1.1 [3] John A. Gubner. Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press, 2006. 4 [4] B. P. Lathi. Modern Digital and Analog Communication Systems. Oxford University Press, 1998. [5] C. Britton Rorabaugh. Communications Formulas and Algorithms: For System Analysis and Design. Mcgraw-Hill, 1990.
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