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Signal & Linear System Analysis Principles of Communications I (Fall, 2002) Signal & Linear System Analysis 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 prob- ability 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(ω0t+θ ) jθ jω0t x(t) = Ae = Ae ⋅ e ~ 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 T 1 0 2 P ≡ | x(t) | dt 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 ex- pansion by orthogonal basis functions is an optimal LSE approxima- tion 2. Is there a very good set of orthogonal basis functions? 3. Concept and relationship of spectrum, bandwidth and infinite continu- ous 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.
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