Signal & Linear System Analysis

Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Signal Model and Classifications z Deterministic vs. Random Deterministic signals: completely specified function of time. Predictable, no uncertainty

e.g. x(t) = Acosω 0t , −∞ < t < ∞ ; where A and ω 0 are constants

Random signals (stochastic signals): take on random values at any given time instant and characterized by pdf (probability density function) “Not completely predictable”, “with uncertainty” e.g. x(n) = dice value shown when tossed at time index n

Model: A (large, maybe ∞ ) set of waveforms each associated with a probability measure e.g. pdf characterizing the out-of-band radio noise

NCTU EE 1 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis z Periodic vs. Aperiodic

Periodic signal: A signal x(t) is periodic iff ∃ a constant T0 , such that

x(t + T0 ) = x(t) , ∀t

The smallest such T0 is called “fundamental period” or “simply period”.

Aperiodic signal: cannot find a finite T0 such that x(t + T0 ) = x(t) , ∀t

z Phasor signal & Spectra A special periodic function ~ j(ω t+θ ) jθ jω t x(t) = Ae 0 = Ae ⋅ e 0

~ jθ x(t) ≡ rotating phasor ; Ae ≡ phasor ; A, θ ≡ real numbers Why this complex signal? 1. Key part of modulation theory 2. Construction signal for almost any signal 3. Easy mathematical analysis for signal 4. Phase carries time delay information

More on Phasor Signal:

1. Information is contained in A and θ (given a fixed f0 (or ω0 )) . 2. The related real sinusoidal function: ~ 1 1 x(t) = Acos(ω t +θ ) = Re{x(t)} or x(t) = Acos(ω t +θ ) = ~x(t) + ~x *(t) 0 0 2 2 3. In vector form graphically:

NCTU EE 2 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

4. Frequency-domain representation

Single-sided (SS) amp. & phasor vs. double-sided (DS):

Line spectra:

E.g. find SS and DS spectra of x(t) = Asin(ω0t +θ ) Ans: put into cosine form first

z Singularity Functions: opposed to regular functions

Unit impulse function δ (t) : 1. Defined by

∞ 0+ 0+ 0+ x(t)δ (t)dt = x(t)δ (t)dt = x(0) δ (t)dt = x(0) ; δ (t)dt = 1 ∫−∞ ∫0 ∫0 ∫0 − − −

2. It defines a precise sample point of x(t) at time t (or t0 if δ (t − t0 ) )

∞ x(t0 ) = x(t)δ (t − t0 )dt ∫−∞

3. Basic function for linearly constructing a time signal ∞ x(t) = x(τ )δ (t −τ )dτ ∫−∞

4. Properties: (Z & T, pp. 25-26 (don’t bother properties 5 and 6)) 1 δ (at) = δ (t) ; δ (t) = δ (−t) : even function | a |

NCTU EE 3 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

5. What is δ (t) precisely? some of intuitive ways of realizing it:

⎧ 1 ⎪lim , | t |< ε E.g. 1 δ (t) = ⎨ε→0 2ε ⎩⎪ 0, otherwise (or elsewhere)

2 ⎛ 1 πt ⎞ E.g. 2 δ (t) = limε⎜ sin ⎟ ε →0 ⎝πt ε ⎠

Unit step function u(t): t du(t) u(t) = δ (λ)dλ; δ (t) = ∫−∞ dt

z Signal types classified by energy & power This classification will be needed for the later analysis of deterministic and random signals

T 2 Energy: E ≡ lim | x(t) | dt T →∞ ∫−T

T 1 2 Power: P ≡ lim | x(t) | dt T →∞ 2T ∫−T

Energy Signals: iff 0 < E < ∞ (P = 0)

Power Signals: iff 0 < P < ∞ (E = ∞)

NCTU EE 4 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

−αt Example-1 x1(t) = Ae u(t)

Example-2 x2 (t) = Au(t)

Example-3 x3 (t) = Acos(ω0t +θ ) Note: 1. If x(t) is periodic, then it is meaningless to find its energy, we only need to check its power

1 T0 P ≡ | x(t) |2dt T0 ∫t+T0 2. Noise is often persistent and is often a power signal 3. Deterministic and aperiodic signals are often energy signals 4. A realizable LTI system can be represented by a signal and mostly is a energy signal 5. Power measure is useful for signal and noise analysis 6. The energy and power classifications of signals are mutually exclusive (cannot be both at the same time). But a signal can be neither energy nor power signal

Signal Space & Orthogonal Basis

z Applying the Sophomore’s Linear Algebra

Basis vectors (for vector space): (essential in DSP & communication theory)

N-dimensional basis vectors: b1,b2 ,L,bN Degree of freedom and independence:

⎡ p⎤ E.g.: In geometry, any 2-D vector x = ⎢ ⎥ can be decomposed into com- ⎣q⎦ ponents along two orthogonal basis vectors, (or expanded by these two vectors)

x = x1b1 + x2 b2 Meaning of “linear” in linear algebra:

x + y = (x1 + y1)b1 + (x2 + y2 )b2

NCTU EE 5 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Basis functions (for function space): (indispensable for general signal analysis) A general function can also be expanded by a set of basis functions (in an approximation sense) ∞ x(t) ≈ ∑ X nφn (t) n=−∞ or more feasibly

N x(t) ≈ ∑ X nφn (t) n=1

Define the matching (or correlation) operation as

∞ ∞ ∞ x(t)φ (t)dt ≈ X φ (t)φ (t)dt ∫ m ∑ n ∫ n m −∞ n=−∞ −∞

If we define orthogonality as ∞ ⎧1, n = m φ (t)φ (t)dt ≡ δ (n − m) ≡ ∫ n m ⎨ −∞ ⎩0, o.w. then ∞ x(t)φ (t)dt = X ∫ m m −∞

E.g. φm (t) = cos(mω0t) for periodic even x(t); set of φm (t) orthogonal? E.g. Taylor’s expansion for cosine function, basis functions? orthogonal?

Remarks: 1. Using Freshmen calculus can show that function approximation expansion by orthogonal basis functions is an optimal LSE approximation 2. Is there a very good set of orthogonal basis functions? 3. Concept and relationship of spectrum, bandwidth and infinite continuous basis functions

NCTU EE 6 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Fourier Series & Fourier Transform

∞ j2πnf0t z Fourier Series: x(t) = ∑ X ne n=−∞

t0 +T0 1 − j2πnf0t X n = x(t)e dt ∫t T0 0

Sinusoidal Representation ∞ ˆ j2πnf0t j2π (−n) f0t x(t) = X 0 + ∑[]X n e + X −n e n=1 If x(t) is real,

j∠X − j∠X j∠X n , −n n X n = X n e X −n = X −n e = X n e

∞ ˆ j(2πnf0t +∠Xn) − j(2πnf0t+∠Xn) x(t) = X 0 + ∑ X n [e + e ] n=1 ∞ = X 0 + ∑2 X n cos(2πnf0t + ∠X n ) Cosine FS n=1 Note: Index starts from 1 (not ∞ )

Trigonometric FS: ∞ ˆ x(t) = X 0 + ∑ X n []cos(∠X n )cos(2πnf0t) − sin(∠X n )sin(2πnf0t) n=1 ∞ ∞ = X 0 + ∑an cos(2πnf0t) + ∑bn sin(2πnf0t) n=1 n=1

where an = 2 X n cos(∠X n ) , bn = 2 X n sin(∠X n )

2 t0 +T0 Or, an = x(t)cos(2πnf0t)dt ∫t 0 T0

2 t0 +T0 bn = x(t) sin(2πnf0t)dt ∫t 0 T0

a ∞ ∞ ˆ 0 x(t) = + ∑∑an cos(2πnf0t) + bn sin(2πnf0t) 2 nn=11=

NCTU EE 7 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Properties of Fourier series

t0 +T0 1 − j2πnf 0t X n = x(t)e dt ∫t 0 λn

t0 +T0 1 − j2π (0) f 0t “DC” coefficient: X 0 = x(t)e dt ∫t 0 T0

1 t0 +T0 = x(t)dt = average value of x(t) ∫t 0 T0

1 t0 +T0 “AC” coefficients: X n = x(t)[cos(2πnf0t) − jsin(2πnf0t)]dt ∫t 0 T0

1 t0 ++T0 1 t0 T0 = x(t)cos(2πnf0t)dt − j x(t)sin(2πnf0t)dt ∫∫t t 0 0 T0 T0

j∠Xn If x(t) is real, then, X n = Re[X n ] + j Im[X n ] = X n e

1 t0 +T0 where Re[X n ] = x(t)cos(2πnf0t)dt ∫t 0 T0

1 t0 +T0 Im[X n ] = x(t)sin(2πnf0t)dt ∫t 0 T0 Hence,

Re[X −n ] = Re[X n ] Im[X −n ] = − Im[X n ] * Æ (even function) and X −n = X n X −n = X n

∠X −n = −∠X n (odd function)

Linearity x(t) ↔ ak

y(t) ↔ bk

Ax(t)+By(t) ↔ Aak + Bbk

Time Reversal

x(t) ↔ ak

x(-t) ↔ a-k

Time Shifting

− jk2πf0t0 x(t − t 0 ) ↔ e a k

NCTU EE 8 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Time Scaling

x(αt) ↔ ak But the fundamental frequency changes

Multiplication

x(t) ↔ ak

y(t) ↔ bk ∞ x(t)y(t) ↔ ∑albk−l l=−∞

Conjugation and Conjugate Symmetry

x(t) ↔ ak

x*(t) ↔ a*-k

If x(t) is real ⇒ a-k = ak*

Parseval’s Theorem Power in time domain = power in frequency domain 2 1 t1 +T0 Px = x(t) dt ∫t1 To

∞ ∞ 1 ⎡ 2 ⎤ 2 Px = ⎢ ∑To X n ⎥ = ∑ X n T0 ⎣n=−∞ ⎦ n=−∞

NCTU EE 9 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Example: half-rectified sinewave

Example:

NCTU EE 10 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Fourier Series Fourier Transform

Good orthogonal basis functions for a Good orthogonal basis functions for a periodic function: aperiodic function: 1. Intuitively, basis functions should 1. Already know sinusoidal func- be also periodic tions are good choice 2. Intuitively, periods of the basis 2. Sinusoidal components should functions should be equal to the not be in a “fundamental & har- period or integer fractions of the monic” relationship target signal 3. Aperiodic signals are mostly fi- 3. Fourier found that sinusoidal nite duration functions are good and smooth 4. Consider the aperiodic function functions to expand a periodic as a special case of periodic func- function tion with infinite period

Synthesis & analysis: (reconstruction & Synthesis & analysis: (reconstruction & projection) projection) Given periodic x(t) with period Given aperiodic x(t) with period 1 1 T = → ∞ T = 0 , ω0 = dω = 2πdf , we can 0 , ω0 = 2πf0 , it can be synthe- df f0 synthesize it as sized as ∞ ∞ ∞ jndωt 1 jωt jnω t x(t) ≈ lim X ne = X (ω)e dω x(t) ≈ X e 0 dω −>0 ∑ 2π ∫ ∑ n n=−∞ −∞ n=−∞ ∞ = X ( f )e j2πft df X n : Spectra coefficient, spectra ∫ amplitude response −∞

To synthesize it we must first analyze it By orthogonality ∞ − j2πft and find out X . X ( f ) = x(t)e dt n ∫−∞ By orthogonality ≡ Fourier Transform of x(t) t0 +T0 1 − jnω0t X n = x(t)e dt ≡ frequency response of x(t) T ∫t0 0 Hence, ∞ x(t) = ∫ X ( f )e j2πft df ≡ Inverse FT −∞

NCTU EE 11 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Frequency components: Frequency components: 1. Decompose a periodic signal into 1. Decompose an aperiodic signal into countable frequency components uncountable frequency components 2. Has a fundamental freq. and many 2. No fundamental freq. and contain all other harmonics possible freq. j 2πft j∠X ( f ) j 2πft X e jnω0t =| X | e j∠X n e jnω0t X ( f )e =| X ( f ) | e e n n | X |: amplitude − ∞ < f < ∞ n ∠X n : phase of | X n | 3. Discrete line spectra 3. Continuous spectral density

Power Spectral Density: Energy Spectral Density: 2 G( f ) ≡| X ( f ) |2 | X n | and and (by Parseval’s equality) ∞ ∞ ∞ 2 2 t0 +T0 E = | x(t) | dt = | X ( f ) | df 1 2 2 ∫−∞ ∫−∞ P = | x(t) | dt = | X n | ∫t ∑ T0 0 n=−∞

In real basis functions: In real basis functions:

As your exercise example! X e jnω0t + X e− jnω0t n −n = 2 | X n | cos(nω0t + ∠X n ) ∞ x(t) = X 0 + ∑ 2 | X n | cos(nω0t + ∠X n ) n=1 ∞ = X 0 + ∑ An cos nω0t n=1 ∞ + ∑ Bn sin nω0t n=1 * note that X −n = X n for real x(t)

NCTU EE 12 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

What x(t) has Fourier series? What x(t) has Fourier Transform?

1. Expansion by orthogonal basis func- 1. Expansion by orthogonal basis functions can be shown is equivalent to tions can be shown is equivalent to X ( f ) finding X n using the LSE (or MSE) finding using the LSE (or cost function: MSE) cost function: T N 2 j2πft 2 2 jnω0t 2 E{[eT (t)] } = E{[x(t) − X ( f )e df ] } E{[eN (t)] } = E{[x(t) − ∑ X ne ] } ∫−T n=−N

2 2 2. Would E{[eN (t)] } → 0 as N → ∞ ? 2. Would E{[eT (t)] } → 0 as T → ∞ ? 3. This requires square integrable condi- 3. This requires square integrable condition (for the power signal): tion (for the energy signal): ∞ 2 2 ∫ | x(t) | dt < ∞ | x(t) | dt < ∞ T0 ∫−∞

and not necessarily| eN (t) |→ 0 4. Dirichlet’s conditions: 4. Dirichlet’s conditions: (a) finite no. of finite discontinuities (a) finite no. of finite discontinuities (b) finite no. of finite max & min. (b) finite no. of finite max & min. (c) absolute integrable (c) absolute integrable ∞ | x(t) | dt < ∞ | x(t) | dt < ∞ ∫T ∫ 0 −∞ Dirichlet’s condition implies conver- Dirichlet’s condition implies conver- gence almost everywhere, except at gence almost everywhere, except at some discontinuities. some discontinuities.

z Some General Symmetry Definitions

(a) Symmetric (even): x(t) = x(−t), x(t) real

(b) Conjugate –symmetric: x(t) = x *(−t)

(d) Conjugate –anti-symmetric: x(t) = −x*(−t)

NCTU EE 13 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis z Symmetry Properties of x(t) and Its Fourier Function

* For real periodic x(t), X −n = X n , or | X −n |=| X n |, ∠X −n = −∠X n .

For real aperiodic x(t), X ( f ) = X * (− f ) or | X ( f ) |=| X (− f ) |, ∠X ( f ) = −∠X (− f )

Example:

z Fourier Transform of Singular Functions δ (t) is not an energy signal (hence doesn’t satisfy Dirichlet condition). However, its FT can be obtained by formal definition.

FT FT δ (t) ⎯⎯→ 1, 1 ⎯⎯→ δ ( f ),

FT − j2πf0 jπf0t FT Aδ (t − t0 ) ⎯⎯→ Ae , Ae ⎯⎯→ Aδ ( f − f0 ),

∞ Example: ∑δ (t − nT0 ) n=−∞ z Fourier Transform of Periodic Signals—Periodic signals are not energy signals (don’t satisfy Dirichlet’s conditions)

∞ ∞ x(t) x(t) = X e jnω0t Given a periodic , ∑ n ⇒ X ( f ) = ∑ X nδ ( f − nf0 ) n=−∞ n=−∞

Example-1 cos 2πf0t

∞ Example-2 ∑δ (t − nT0 ) n=−∞

NCTU EE 14 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis z Properties of Fourier Transform (p.223 of O.W.Y)

NCTU EE 15 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis z Fourier Transform Pairs (p.225 of O.W.Y)

NCTU EE 16 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

z Relationship Between FT of an Aperiodic Signals & Its Periodic Extension Let FT of an aperiodic pulse signal p(t) be ℑ{p(t)} = P( f ) ,

We can generate a periodic signal x(t) by duplicating p(t) at every Ts in- terval , then ∞ ∞ x(t) = [ ∑δ (t − nTs )]* p(t) = ∑ p(t − nTs ) n=−∞ n=−∞ From convolution theorem,

∞ X ( f ) = ℑ{[ ∑δ (t − nTs )]}× P( f ) n=−∞ ∞ ∞ = f s ∑δ ( f − nf s )× P( f ) = ∑ f s P(nf s )δ ( f − nf s ) n=−∞ n=−∞

NCTU EE 17 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Poisson sum formula: By taking inverse FT of above eq.

∞ ∞ −1 −1 ℑ {X ( f )} = x(t) = ∑ p(t − nTs ) = ℑ { ∑ f s P(nf s )δ ( f − nf s )} n=−∞ n=−∞ ∞ ∞ −1 j2πnfst = ∑ f s P(nf s )ℑ {δ ( f − nf s )} = ∑ f s P(nf s )e n=−∞ n=−∞

∞ ∞ j2πnf s t ∑ p(t − nTs ) = ∑ f s P(nf s )e --> Poisson sum formula n =−∞ n =−∞

Power Spectral Density & Correlation z Energy Signal φ(τ ) ≡ ℑ−1{G( f )} = ℑ−1[X ( f )X * ( f )] = ℑ−1[X ( f )]∗ ℑ−1[X * ( f )] ∞ T = x(τ ) ∗ x(−τ ) = x(λ)x(λ + τ )dλ = lim x(λ)x(λ +τ )dλ ∫−∞ T →∞ ∫−T Time-average autocorrelation function

φ(0) = E signal energy

z Time average autocorrelation function of power signals

R(τ ) ≡ x(τ )x*(t + τ )

T ⎧ 1 * ⎪ lim x(t)x (t + τ )dt, ⎪T→∞ 2T ∫−T if aperiodic power signal = ⎨ 1 ⎪ ∫ x(t)x*(t + τ )dt, if periodic power signal ⎩⎪ T0 T0

∞ R(0) = S( f )df ∫−∞

S( f ) = ℑ{R(τ )} Power spectral density

For periodic power signal ∞ 2 S( f ) = ℑ{R(τ )} = ∑| X n | δ ( f − nf0 ) n=−∞

NCTU EE 18 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Interpretation

1. φ(τ ) and R(τ ) measure the similarity between the signal at time t and t + τ

2. G( f ) and S( f ) represents the signal energy or power per unit frequency at freq. f.

Properties of R(τ ) :

2 1. R(0) = power = x (t) ≥ R(τ ) , ∀τ , max{R(τ )} = R(0)

* 2. R(τ ) is even for real signal: R(−τ ) ≡ x(t)x (t −τ ) = R(τ ) 3. If x(t) does not contain a periodic component

2 lim R(τ ) = x(t) |τ|→∞

4. If x(t) is periodic with period T0 , then R(τ ) is periodic in τ with the same period

5. S( f ) = ℑ{R(τ )} ≥ 0, ∀f

z Crosscorrelation of two power signals x(t), y(t)

* * Rxy (τ ) ≡ x(t)y (t −τ ) = x(t + τ )y (t) 1 T = lim x(t)y*(t −τ )dt T→∞ 2T ∫−T

z Crosscorrelation of two energy signals x(t), y(t)

∞ * φxy (τ ) ≡ x(t)y (t −τ )dt ∫−∞

* * Remarks: Rxy (τ ) = Ryx (−τ ) , φxy (τ ) = φ yx (−τ )

NCTU EE 19 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Signals & Linear Systems

x(t) y(t) Η 1. y(t) = H{x(t)} , 2. H: an operator representing some mechanism and operation on x(t) and/or y(t) to produce y(t); generally it can be approximated by a differential equation. 3. More specifically, the linear constant-coefficient differential eq. suits for most applications and makes life a lot easier!

z Linear & Time-Invariant (LTI) System Linear systems: Satisfies superposition principle

y(t) = H{α1x1(t) + α2 x2 (t)} = H{α1x1(t)} + H{α2 x2 (t)} = y1(t) + y2 (t)

Time-invariant: Delayed input produces a delayed output (due to the non-delayed input)

y(t) = H{x(t)} --> H{x(t − t0 )} = y(t − t0 )

Complete characterization of LTI systems: The unit impulse function is key to the characterization.

h(t) ≡ H{δ (t)}

∞ x(t) = x(λ)δ (t − λ)dλ ∫−∞

∞ ∞ y(t) = H{x(t)} = H{ x(λ)δ (t − λ)dλ} = x(λ)H{δ (t − λ)}dλ ∫−∞ ∫−∞

NCTU EE 20 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

If TI,

∞ y(t) = H{x(t)} = x(λ)h(t − λ)dλ ≡ x(t) * h(t) ∫−∞

∞ H{x(t − t0 )} = x(λ)h(t − t0 − λ)dλ = y(t − t0 ) ∫−∞

NCTU EE 21 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Convolution form holds iff LTI

1. Duality of signal x(t) & system h(t)

∞ ∞ y(t) = x(λ)h(t − λ)dλ = h(λ)x(t − λ)dλ ∫−∞ ∫−∞

Convolution theorem:

∞ ℑ{y(t)} = Y ( f )ℑ{ h(λ)x(t − λ)dλ} = H ( f )X ( f ) ∫−∞

Key application: generally H ( f )X ( f ) is easier than x(t) * h(t) ….

BIBO stability:

A system is BIBO if output is bounded, given any bounded input ∞ ∞ max{| y(t) |} = max{| h(λ)x(t − λ)dλ |} = max{| x(t) |} | h(λ) |dλ < ∞ ∫−∞ ∫−∞ ∞ ⇒ | h(λ) |dλ < ∞ ⇒ main element of Dirichlet condition ∫−∞

Causality: 1. A system is causal if: current output does not depend on future input; or current input does not contribute to the output in the past Æ A truth in reality. 2. All the systems in nature world are causal. Causality generally is not a problem in most circumstance. However, it matters frequently when we want to counteract the effect of hostile communication channel, such that we need an ideal equalizer, which is non-causal.

∞ ∞ y(t) = h(λ)x(t − λ)dλ = h(λ)x(t − λ)dλ ∫−∞ ∫0

⇒ h(t) = 0, for t < 0

NCTU EE 22 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Paley-Wiener Condition:

∞ 2 If (1) H( f ) df < ∞ ∫−∞

(2) h(t) = 0, t < ∞

∞ ln H(f) ⇒ df < ∞ ∫−∞ 1 + f 2 Remarks: |H(f)| cannot grow too fast. |H(f)| cannot be exactly zero over a finite band of frequency

∞ 2 If (1) H( f ) df < ∞ ∫−∞

∞ ln H(f) (2) df < ∞ ∫−∞ 1 + f 2

⇒ ∃ ∠H(f) such that H(f) is causal (h(t) = 0, t < ∞)

z Eigenfunctions of LTI System

If Η{g(t)} = αg(t) , where α is a constant w.r.t time parameter, then

⎧ α : eigenvalue ⎨ ⎩g(t) : eigen function

st x(t) = Ae , s : an arbitrary complex number

∞ ∞ y(t) = h(λ)Aes(t−λ)dλ = [ h(λ)e−sλ dλ]Aest = αx(t) ∫−∞ ∫−∞ ∞ α = h(λ)Ae−sλ dλ ∫−∞

∞ − j2πfiλ j2πfit s = j2πf t y(t) = [ h(λ)e dλ]Ae = αx(t) Let i , then ∫−∞

NCTU EE 23 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

z Correlation functions related by LTI systems:

∞ 1. Ryx (τ ) = h(τ )∗ Rx (τ ) = h(λ)Rx (τ − λ)dλ ∫−∞

* 2. Ry (τ ) = h (−τ )∗h(τ )∗Rx (τ )

3. S yx ( f ) = H ( f )⋅ S x ( f )

2 4. S y ( f ) =| H ( f ) | ⋅S x ( f )

pf) (1) & (2) from definitions. (3) & (4) from ℑ{h(−τ )} = H (− f ) and

* * ℑ{h (τ )} = H (− f )

z System Transmission Distortion & System Frequency Response (a) Since almost any input x(t) can be represented by a linear combination of

orthogonal sinusoidal basis functions e j2πft , we only need to input

j2πft Ae to the system to characterize the system’s properties, and the eigenvalue

∞ − j2πft α = h(t)e dt = H ( f ) ∫−∞

carries all the system information responding to Ae j2πft . (b) In communication theory, transmission distortion is of primary concern in high-quality transmission of data. Hence, the system representing transmission channel is the key investigation target.

Three types of distortion of a transmission channel: 1. Amplitude distortion: linear system but the amplitude response is not constant. 2. Phase (delay) distortion: linear system but the phase shift is not a linear function of frequency. 3. Nonlinear distortion: nonlinear system

NCTU EE 24 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Group delay: 1 dθ ( f ) Tg ( f ) = − , θ ( f ) = ∠H ( f ) 2π df

For a linear phase system, θ ( f ) = ∠H 0 − 2πft0

Tg ( f ) = t0 , a constant

If Tg(f) is not a constant, sinusoidal inputs of different frequencies have different delays. θ ( f ) T ( f ) = − Cf. Phase delay: p 2πf

Ideal general filters:

NCTU EE 25 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Realizable filters approximating the ideal filters: 1. Butterworth filter: simple 2. Chebyshev filter: smaller maximum deviation 3. Bessel filter: approximately linear phase in passband

NCTU EE 26 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

z Time-Bandwidth Product—uncertainty principle It can be argued that a narrow time signal has a wide (frequency) bandwidth, and vice versa: (duration) ×(bandwidth) ≥ constant

NCTU EE 27 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Sampling Theory Ideal sampling Ideal sampling signal: impulse train (an analog signal) ∞ s()t = ∑δ (t − nT ), T: sampling period n=−∞ Analog (continuous-time) signal: x(t)

Sampled (continuous-time) signal: xδ (t)

∞ xδ ()t = x ()()t s t = x ()t ∑δ (t − nTs ) n=−∞ ∞ ∞ = ∑ x()()()()t δ t − nTs = ∑ x nTs δ t − nTs n=−∞ n=−∞

∞ X δ ()f = X ( f )∗ S ()f = X ()f ∗[ f s ∑δ (f − kf s )] k =−∞ ∞ ∞ = f s ∑∑X ()f ∗δ (f − kf s )= f s X (f − kf s ) kk=−∞ =−∞ Two Cases:

(1) no aliasing: f s > 2W , and

(2) aliasing: f s < 2W , where W is the highest nonzero frequency component of X ( f ) . After sampling, the replicas of X ( f ) overlap in frequency domain. That is, the higher frequency components of X ( f ) overlap with the

lower frequency components of X ( f − f s ).

NCTU EE 28 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

Nyquist Sampling Theorem:

Let x(t) be a bandlimited signal with X ( f ) = 0 for | f |≥ W . (i.e., no com-

ponents at frequencies greater than W ) Then x(t) is uniquely determined 1 by its samples x[n] = x(nTs ), n = 0,±1,±2,K, if f s = ≥ 2W . Ts

-- Undersampling: f s < 2W

-- Overdampling: f s > 2W

Reconstruction: Ideal reconstruction filter: f H ( f ) = H Π( )e− j2πft0 , W ≤ B ≤ f −W 0 2B s

− j2πft0 ⇒ Y ( f ) = f s H 0 X ( f )e

⇒ y(t) = f s H 0 x(t − t0 )

Alternative expression

∞ ∞ y(t) = ∑ x(nTs )h(t − nTs ) = 2BH 0 ∑ x(nTs )sinc[2B(t − t0 − nTs )] n=−∞ n=−∞

Two types of reconstruction errors

NCTU EE 29 Principles of Communications I (Fall, 2002) Signal & Linear System Analysis

DFT & FFT

z DFT

2πnk N −1 j 1 N xn = ∑ X k e , n = 0,1,L, N − 1 N k=0 2πnk N −1 − j N X k = ∑ xne , k = 0,1,L, N −1 n=0

jω H (e ) = H (z) |z=e jω N −1 N −1 jω − j2πnk / N nk H (k) = H (e ) | 2πk = h(n)e = h(n)WN ω = ∑ ∑ N n=0 n=0

z Fast convolution by FFT

N −1 N −1 1 nk 1 nk y(n) = ∑Y (k)WN = ∑ H (k)X (k)WN N k =0 N k=0

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