Signals & Linear Systems Analysis Chapter 2&3, Part II

Transcription

1 Signals & Linear Systems Analysis Chapter &3, Part II Dr. Yun Q. Shi Dept o Electrical & Computer Engr. New Jersey Institute o echnology shi@njit.edu et used or the course: <Modern Digital and Analog Communication Systems>, 4 th Edition, Lathi and Ding, Oord Dr. Shi Digital Communications Signal and System Signal: A signal is deined as the time history o some quantity, usually a voltage or current. Deterministic Signals Random Signals System: A system is a combination o devices and networks subsystems chosen to perorm a desired unction. Dr. Shi Digital Communications

2 Signal Models Deterministic Signals are modeled as completely speciied unctions o time. Eamples: Acosω t, < t < Sinusoidal A: magnitude : angular requency ω t < otherwise unit rectangular pulse unction denoted by t Dr. Shi Digital Communications 3 Signal Models Random signals are signals that take on random values at any given time instant and must be modeled probabilistically. Eample Figure in the net slide a A sinusoidal signal deterministic b Unit rectangular pulse signal deterministic c A random signal its one sample unction Dr. Shi Digital Communications 4

3 Dr. Shi Digital Communications 5 Signal Models Periodic signals A signal is periodic i t +, < t < where the constant is a period. deterministic Fundamental period he smallest period is reerred to as undamental period. Aperiodic signals Any signal not satisying t + t is called aperiodic. Dr. Shi Digital Communications 6 3

4 Signal Models Rotating phasor A useul tool to deal with sinusoidal quantities. A rotating phasor: hree parameters: ~ j ω t + θ Ae A: amplitude θ: phase in radians : requency in radians per sec ω jθ jω t Phasor: Ae where is implicit. e < t < Dr. Shi Digital Communications 7 he rotating phasor is periodic: ~ ~ t +, π ω ~ j Ae Q t + [ ω t + +θ ] π π Acos ω t + + θ + jasin ω t + + θ ω ω A[ cos ωt + θ + j sin ωt + θ ] j ω +θ Ae ~ Dr. Shi Digital Communications 8 4

5 Power and Energy p : power Signal Classiications: Energy and Power It is normal power with Ω impedance. Higher-energy signals are detected more reliably with ewer errors than lower energy signals. Dr. Shi Digital Communications 9 Signal s Energy and Power: Deinition Let be an arbitrary signal possibly comple unction. Its total energy is: E lim dt dt Its average power is: P lim dt Dr. Shi Digital Communications 5

6 Energy signal vs. Power signal he unction is an energy signal i < E < Hence P, because o having non-zero and inite energy. he unction is a power signal i hus E < P <, because o having non-zero and inite power. Dr. Shi Digital Communications For a periodic signal p t + P p t dt t is the period. No need to carry out the limiting operation to ind P or a periodic signal. Energy and power classiications are mutually eclusive.. An energy signal must have zero average power, not a power signal.. A power signal must have ininite energy, not an energy signal here are some signals: neither energy nor power signals. he ramp signal is such an eample. Dr. Shi Digital Communications 6

7 Eample Reer to a igure about unit step unction. αt Ae u A E lim dt dt α t e αt e A α t A A e α α α is an energy signal. α > u t t < Dr. Shi Digital Communications 3 I α, Au, ininite energy. P lim dt lim A dt A Final power. It is a power signal Dr. Shi Digital Communications 4 7

8 Eample Reer to igure shown beore. ω θ p P Acos A + t t A t + cos ω t + θ dt A [ + cos ωt + θ ] dt It is a power signal. a sinusoidal signal ininite energy Dr. Shi Digital Communications 5 A requently used skill: cos π Q ω [ ω t + θ ] dt π π ω π hat is, period o cos[ ω t + θ ] becomes hal. Integration o a sinusoid within an integral number o periods is zero. Dr. Shi Digital Communications 6 8

9 Energy Spectral Density ESD. Derivation otal Energy: E e dt * jπt * e d jπt dt dt d Dr. Shi Digital Communications 7 * d ξ d. Deinition ESD Energy spectral density o a signal. ξ Ε ξ d ξ d Dr. Shi Digital Communications 8 9

10 3. Unit ESD t e jπt dt : volts seconds Q On a per ohm basis or total energy & average power volts volts sec onds sec onds ohm watts sec ond, sec ond hertz hertz joules / hertz It is the Energy Density. Dr. Shi Digital Communications 9 4. Comment. Parseval s heorem Rayleigh s Energy heorem E dt d Comment. is energy spectral density. ξ Q ξ d E Dr. Shi Digital Communications

11 Power Spectral Density PSD.Consider as a real-valued power signal, then its average power is P lim t dt his is similar to deinition given beore slide ecept that the absolute value sign has been removed. Dr. Shi Digital Communications PSD. I is a periodic signal with period, then its average power is where n P P n n dt * dt is Fourier series FS coeicient o. Dr. Shi Digital Communications

12 PSD his can be proved by using FS o. Loosely speaking, n n ep jπnt jπn ' t n ep n ' All cross-product terms, ecept as n n, due to orthogonality o comple eponential unction, thus leading to n. n Dr. Shi Digital Communications 3 PSD 3. Deine PSD o a periodic unction as: n PSD n δ n Recall that a periodic unction has line spectra. nδ n n Dr. Shi Digital Communications 4

13 hen P n n PSD PSD d n n n n δ n δ n d d Hence, PSD thus deined is, indeed, PSD. Dr. Shi Digital Communications 5 PSD 4. For a non-periodic unction, we irst truncate as in then ind its F:., It can be shown that PSD lim Dr. Shi Digital Communications 6 3

14 ESD: PSD: PSD PSD Summary: ESD and PSD ξ lim n δ n n,, periodic non periodic F{ } n FS coeicient o F{ } Dr. Shi Digital Communications 7 Autocorrelation Correlation unction Another approach to signal and systems. Autocorrelation A measure o similarity matching o a signal with its delayed version. Dr. Shi Digital Communications 8 4

15 Deinition Autocorrelation o an Energy Signal R τ t + τ dt, < τ < A measure o how closely the signal matches a copy o itsel as the copy is shited τ units in time he larger the more correlated. Dr. Shi Digital Communications 9 Properties: Autocorrelation o an Energy Signal. R τ R -τ symmetrical in τ about τ. R τ R τ maimum value o Rτ occurs at τ F R τ ξ ESD 3. an important F pair t dt E R 4. energy Dr. Shi Digital Communications 3 5

16 Autocorrelation o a Power Signal Deinition R τ lim or t + τ t + τ dt, < τ < For a periodic signal with period R τ t + τ dt < τ <, Dr. Shi Digital Communications 3 Properties R τ o a real-valued periodic signal :. R τ R -τ even symmetry. R τ R, τ maimum value F R τ PSD 3. an important F pair R 4. average power dt Dr. Shi Digital Communications 3 6

17 Random Signals Random eperiment: E Outcome o r.e.: ξ Sample space: S Deinition o probability Aioms o probability Random event: A Random variable r.v.: A a unction o A, a real number, numerical attribute Dr. Shi Digital Communications 33 Random Signals CDF CDF Cumulative Distribution Function F P Properties:. F. F F, i 3. F 4. F + 5. continuous rom the right F Dr. Shi Digital Communications 34 7

18 8 Dr. Shi Digital Communications 35 Random Signals PDF PDF Properties.. d F F P P P d df F F d Dr. Shi Digital Communications 36 For discrete r.v. s, instead o, we oten use P i probability mass unction PMF Strictly speaking < i i i i i i i u P F i P ; δ pd o discrete r.v. cd o discrete r.v.

19 Ensemble Average statistical average Numerical attributes o r.v. Mean value, m epected value m n th moment E d E n n d n, mean value o, mean n, mean-square value o n 3, n 4, } very important this and net slides Dr. Shi Digital Communications 37 Ensemble Average statistical average Central moment E n [ m ] n m d st central moment: nd central moment: E [ m ] var σ Important ormula: σ [ ] variance E m 3 rd central moment: skewness o pd 4 th central moment: kurtosis o pd Dr. Shi Digital Communications 38 9

20 Random Process r.p. A, A: random event t: time a larger set a collection o r.v. s or, a collection o random sample unction oten simpliied as Dr. Shi Digital Communications 39 Figure 8.- rom <Probability, Random Processes, and Estimation heory or Engineers> by Stark & Woods 994 Digital Communications 4

21 For a speciied event A j, A j, j, a sample unction For a speciied time t k, A, t k k, a r.v. For speciic A j, t k, A j, t k a real value Dr. Shi Digital Communications 4 Statistical Average o a r.p. A r.p. should be completely characterized by pd o all r.v. s. his is diicult. Hence, numerical characterization used oten. A partial description: { mean unction st order autocorrelation unction nd order E[ t k ] d m tk k R t, t E[ t] Dr. Shi Digital Communications 4

22 Stationarity S.S.S. Strict Sense Stationary A r.p. is SSS i none o its statistics are aected by a shit in the time origin. W.S.S.Wide Sense Stationary A r.p. is WSS i its mean and autocorrelation unction do not vary with a shit in the time origin. Dr. Shi Digital Communications 43 Stationarity For a W.S.S. random process, E[ ] m R t, t R t t Relation between SSS and WSS SSS not n essarily a constant only a unction o W SS t t Dr. Shi Digital Communications 44

23 Autocorrelation o a WSS r.p. Let τ t t R τ E [ t t + τ ], < τ < R τ gives us an idea o the requency response I R τ changes slowly as τ changes, then has more low requency components; otherwise, it has more high requency comp. Four similar properties o R τ to that o autocorrelation o an energy power signal. Dr. Shi Digital Communications 45 ime Average & Ergodicity o compute m and R τ needs to have or all, sometimes not possible, preer the time average ime average over a single sample unction o the r.p.: lim t dt always t + τ lim t t + dt τ possible Dr. Shi Digital Communications 46 3

24 An ergodic r.p.: m R τ t +τ ergodicity not nessarily SSS A r.p. is ergodic in the mean, i m A r.p. is ergodic in the autocorrelation unction i τ t + τ R Dr. Shi Digital Communications 47 A reasonable assumption in the analysis o most communication signals in the absence o transient eects is that random waveorms are ergodic in the mean and autocorrelation unction For ergodic r.p., time averages statistical ensemble averages Dr. Shi Digital Communications 48 4

25 some observations: m dc levels o the signal m normalized power o dc component E [ ] total average normalized power E[ ] root-mean-square rms value o σ average normalized power in ac component I m, σ mean-square value o or total power σ rms o ac component I m, σ rms o the signal Dr. Shi Digital Communications 49 PSD o a Random Process A r.p. can generally be considered as a power signal. PSD : PSD lim t, Useul in communication systems: Indicates the distribution o a signal s power in requency domain. Dr. Shi Digital Communications 5 as 5

26 PSD o a Random Process Properties: PSD PSD R P τ PSD PSD F PSD d nonnegative real-valued i is real-valued Dr. Shi Digital Communications 5 Noise in Communication Systems Noise: Unwanted electrical signals always present in electrical system Man-made Noise: switching transients, spark-plug ignition Natural Noise: thermal always eists Gaussian noise central limit theorem slide 54 Dr. Shi Digital Communications 5 6

27 Normalized Gaussian pd m σ n ep σ π n σ Dr. Shi Digital Communications 53 Gaussian Noise A random signal z a + n a: deterministic component o the signal n: additive random noise z a z ep σ π σ Gaussian distribution is etremely important in practice due to the central limit theorem he theorem: he probability distribution o the sum o j statistically independent r.v. s approaches the Gaussian distribution as j, no matter what the individual distribution unctions may be. Dr. Shi Digital Communications 54 7

28 De. White Noise White comes rom the act that white light contains equal amounts o all requencies within the visible band o electromagnetic radiation Dr. Shi Digital Communications 55 Autocorrelation unction N N Rn τ F { PSD } F δ τ n is totally decorrelated rom its time-shited version or any τ > Meaning: Any two dierent samples o a white noise are uncorrelated no matter how close together in time they are taken. Average power P n Dr. Shi Digital Communications 56 N d 8

29 In practice, as long as the bandwidth o the noise is appreciably larger than that o the system, the noise can be considered to have an ininite bandwidth. hermal noise: Additive white Gaussian Noise AWGN Very important in both theory and practice. Dr. Shi Digital Communications 57 Signal ransmission hrough Linear Systems input LI Linear Network output, h, y --- time domain, H, Y --- requency domain h: unit impulse response o the LI system H: transer unction Dr. Shi Digital Communications 58 9

30 h: unit impulse response h y when δ y * h Causality A system is causal i there is no output prior to the time, t, when the input is applied. A system is causal i h, τ h t τ dτ t τ h τ dτ t < y t τ h t τ d τ Dr. Shi Digital Communications 59 H: Frequency transer unction. H F{h} { transer unction. requency response Y F{y} F{ * h} H H Y Dr. Shi Digital Communications 6 3

31 Frequency Response H : Comple in general H H e jθ Amplitude requency response: Phase requency response: θ H Dr. Shi Digital Communications 6 he transer unction o a LI system can be measured by using a sinusoidal testing signal that is swept over the requency o interes since the spectrum o sinusoid is a line at the testing requency. For sinusoidal input, i. e., Acos[ π t + φ ] [ π t + φ + θ ] y A H cos 443 magnitude same { requency new phase Dr. Shi Digital Communications 6 3

32 Random Process & Linear Systems I : a r.p. h: LI system then y: output, another r.p. i.e., every sample unction o the input r.p. a corresponding sample unction PSD : PSD o PSD y : PSD o y PSD y PSD H Dr. Shi Digital Communications 63 he relation between the power spectral density PSD at the input, PSD, and the output, PSD y PSD H PSD y he power transer unction o the LI system PSDy Gh H PSD Q PSDy lim Y PSD lim Dr. Shi Digital Communications 64 Y H 3

33 I is a Gaussian r.p., h is LI, then y is also Gaussian I the input to a LI system is periodic with spectrum given by line spectrum δ n n n where { n }is the comple eponential FS coeicients o the input signal, then the output signal s spectrum is nh n δ n n Y Dr. Shi Digital Communications 65 RC Low-Pass Filter Eample 3- R i + y dy dy Qi C RC + y dt dt From the F property o the dierentiation, RC jπ Y + Y Dr. Shi Digital Communications 66 33

34 34 Dr. Shi Digital Communications 67 < t t e t h RC j RC j C j R C j H or RC j Y H t τ τ π ω ω ω π with τ RC: time constant according to able 3. RC H G h π, + power transer unction Dr. Shi Digital Communications 68 Distortionless ransmission What is an ideal transmission line? In time domain Some time delay is allowed y vs. A scale change in magnitude is allowed y K t-t K: scale change t : time delay

35 In requency domain Y i. e., H K K e j π t H K constant magnitude change θ H π t θ t π t needs to be ied. Phase shit must be proportional to requency in order or the time delay o all components to be identical, i.e., phase _ delay _ θ Equalization : phase or amplitude correction network e j π t Dr. Shi Digital Communications 69 Ideal Filters One cannot build the ideal network described above since it implies an ininite bandwidth capability Sklar s, page 33. An approimation to the ideal ininite-bandwidth network is to use a truncating network that passes all req. components between l and u without distortion, where l and u are the lower and upper cuto requency, respectively. Ideal BPF LPF HPF Dr. Shi Digital Communications 7 35

36 36 Dr. Shi Digital Communications 7 Ideal Filters Fig.. rom <Digital Communications> by Sklar,988 Dr. Shi Digital Communications 7 ake a look at ILPF t j j u u j e e or or H e H H π θ θ < Why more emphasize LPF?

37 37 Dr. Shi Digital Communications 73 [ ] ] [ t t Sinc t t t t Sin u u u u u π π π u u u u t t j t j t j t j d e d e e d e H H F t h } { π π π π Dr. Shi Digital Communications 74 Unit Impulse Response o ILPF

38 Eect o an ILPF on White Noise Eample.. N PSDn PSDY? RY τ? Solution: PSD PSD H Y N n or < otherwise u Dr. Shi Digital Communications 75 R τ F Y { PSD } Y N u sinc[ ] N sinc[ ] { u τ u π uτ 443 area_ o _ PSD width_o _ PSDY, : Y st zero crossing u Ater LPF, is it white noise or not? Not a white noise anymore Dr. Shi Digital Communications 76 38

39 RC LPF: Realizable Filters RC ilter requency analysis o sinusoidal circuits V V out input R + j π c j π c + j π Rc Dr. Shi Digital Communications 77 H θ tan + πrc πrc Dr. Shi Digital Communications 78 39

40 Consider R, i.e., the normalized case. P V V V V V R V.77 V V P P P, P hal power No. o db P log log.3 3dB P Hence, hal-power -3dB Dr. Shi Digital Communications 79 Eect o an RC ilter on white noise Eample.3 G n G G H y R τ F y N n N + πrc N { Gy } τ ep 4Rc Rc eponential _ unction When input is white noise, output o the RC ilter: Not white noise anymore Dr. Shi Digital Communications 8 4

41 Several Useul Realizable Filters Butterworth ilter Chebychev ilter most lat one in passband Ripple in passband smaller variation in stopband For pass-band and stop-band reer to the igure on slide 8 Dr. Shi Digital Communications 8 Butterworth Low Pass Filter H u n, or n, H n + c u ILPF, n n called corner requency Dr. Shi Digital Communications 8 4

42 Butterworth Amplitude Response Dr. Shi Digital Communications 83 Signals, Circuits & Spectra Input signal, its spectrum rect sinc e.g. Circuit, RC circuit, H, θ Output y, Y Case : Output bandwidth is constrained by input signal bandwidth, i.e., H is wideband. Case : Output bandwidth is constrained by ilter bandwidth, i.e., H is narrowband. Dr. Shi Digital Communications 84 4

43 Fig..4, rom <Digital Communications> by Sklar,988 Dr. Shi Digital Communications 85 Bandwidth o Digital Data Baseband vs. Bandpass Double sideband DSB modulation signal low-pass or baseband signal spectrum, - m : baseband bandwidth DSB modulated signal baseband signal cosπ, c c cosπ c Dr. Shi Digital Communications 86 c c m DSB modulated signal local oscillator LO carrier 43

44 44 Dr. Shi Digital Communications 87 [ ] [ ] } {cos c c c c c c t F δ δ π Q and convolution o a normal actor with a δ unction.