SJTU通信原理概論課件2

1、Introduction to Communication PrinciplesCourse Name: Introduction to Communication PrinciplesCourse Code: EI211Course Instructor: Dianguang MaText: Digital and Analog Communication Systems, 6th ed. by Leon W. Couch II, ISBN 0-13-081223-4Contents: Chapters 1 4.CHAPTER 2SIGNALS AND SPECTRAChapter Obje

2、ctivesBasic signal properties (dc, rms, dBm, and power)Fourier transform and spectraLinear systems and linear distortionBandlimited signals and samplingDiscrete Fourier transformBandwidth of signals2-1 PROPERTIES OF SIGNALS AND NOISEIn communication systems, the received waveform is usually categori

3、zed into the desired part containing the information and the extraneous or undesired part. The desired part is called the signal, and the undesired part is called noise.2-1 PROPERTIES OF SIGNALS AND NOISEPractical waveforms that are physically realizable (i.e., measurable in a laboratory) satisfy se

4、veral conditions:The waveform has significant (nonzero) values over a composite time interval that is finite.The spectrum of the waveform has significant values over a composite frequency interval that is finite.The waveform is a continuous function of time.The waveform has a finite peak value.The w

5、aveform has only real values. That is, at any time, it cannot have a complex value a + jb, where b is nonzero.Mathematical models that violate some or all of the conditions listed above are often used, and for one main reason to simplify the mathematical analysis.2-1 PROPERTIES OF SIGNALS AND NOISE2

6、-1 PROPERTIES OF SIGNALS AND NOISEThe time average operator is given byIf the waveform involved is periodic with period T0, the time average operator can be reduced to2-1 PROPERTIES OF SIGNALS AND NOISEThe dc (“direct current”) value of a waveform w(t) is given by its time average, .For any physical

7、 waveform, we are actually interested in evaluating the dc value only over a finite interval of interest, say, from t1 to t2, so that the dc value is2-1 PROPERTIES OF SIGNALS AND NOISELet v(t) denote the voltage across a set of circuit terminals, and let i(t) denote the current into the terminal. Th

8、e instantaneous power associated with the circuit is given byThe average power isThe root-mean-square (rms) value of w(t) isIf the load is resistive, the average power is2-1 PROPERTIES OF SIGNALS AND NOISEThe concept of normalized power is often used by communication engineers. In this concept, R is

9、 assumed to be 1 , although it may be another value in the actual circuit. The average normalized power isw(t) is a power waveform if and only if the normalized average power P is finite and nonzero.The total normalized energy isw(t) is an energy waveform if and only if the total normalized energy E

10、 is finite and nonzero.2-1 PROPERTIES OF SIGNALS AND NOISEPhysically realizable waveforms are of the energy type.Basically, mathematical models are of the power type.Mathematical functions can be found that have both infinite energy and infinite power, e.g., w(t) = e -t.The decibel signal-to-noise r

11、atio isThe decibel power level with respect to 1 milliwatt is2-2 FOURIER TRANSFORM AND SPECTRAThe Fourier transform of a waveform w(t) isW(f) is also called a two-sided spectrum of w(t).All physical waveforms encountered in engineering practice are Fourier transformable. The sufficient condition for

12、 the existence of the Fourier transform is E .The time waveform may be calculated from the spectrum by using the inverse Fourier transform2-2 FOURIER TRANSFORM AND SPECTRAExample 2-4 (VOLTAGE/CURRENT) SPECTRUM OF A SINUSOID2-2 FOURIER TRANSFORM AND SPECTRA2-3 POWER SPECTRAL DENSITY AND AUTOCORRELATI

13、ON FUNCTIONThe normalized power of a waveform will now be related to its frequency domain description by the use of a function called the power spectral density (PSD). The PSD for a deterministic power waveform is2-3 POWER SPECTRAL DENSITY AND AUTOCORRELATION FUNCTIONThe autocorrelation of a real (p

14、hysical) waveform isThe PSD and the autocorrelation function are Fourier transform pairs; that isThe total average normalized power for the waveform w(t) can be evaluated by using any of the four techniques embedded in the following equation:2-3 POWER SPECTRAL DENSITY AND AUTOCORRELATION FUNCTION2-3

15、 POWER SPECTRAL DENSITY AND AUTOCORRELATION FUNCTIONExample 2-10 PSD OF A SINUSOID2-3 POWER SPECTRAL DENSITY AND AUTOCORRELATION FUNCTION2-4 ORTHOGONAL SERIES REPRSENTATION OF SIGNALS AND NOISEHow do we represent the signal or noise waveform?Direct approachTo represent the waveform by the use of a T

16、aylor seriesOrthogonal series representation, such as Fourier series, sampling function seriesFunctions n(t) and m(t) are said to be orthogonal with respect to each other over the interval a t b if they satisfy the conditionIf the functions in the set n(t) are orthogonal, then they also satisfy the

17、relation2-4 ORTHOGONAL SERIES REPRSENTATION OF SIGNALS AND NOISEAssume that w(t) represents some practical waveform, it can be represented over the interval (a, b) by the serieswhere the orthogonal coefficients are given by and the range of n is over the integer values that correspond to the subscri

18、pts that were used to denote the orthogonal functions in the complete orthogonal set.It can be shown that many useful sets are complete, such as the complex exponentials, the harmonic sinusoidal functions, Bessel functions, Legendre polynomials, and the sampling functions.2-5 FOURIER SERIESA physica

19、l waveform may be represented over the interval (a, a+T0) by the complex exponential Fourier serieswhere the complex Fourier coefficients are and where 0 = 2f 0 = 2/T0.If the waveform w(t) is periodic with period T0, the Fourier series representation is valid over all time, because w(t) and n(t) are

20、 periodic with the same fundamental period T0. For this case of periodic waveforms, the choice of a value for the parameter a is arbitrary and is usually taken to be a = 0 or a = - T0 /2 for mathematical convenience.2-5 FOURIER SERIESIf a waveform is periodic with period T0, the (two-sided) spectrum

21、 of the waveform w(t) iswhere f0 = 1/T0 and cn are the complex Fourier coefficients of the waveform.2-5 FOURIER SERIESIf w(t) is periodic function with period T0 and is represented bywherethen the Fourier coefficients are given bywhere H(f) = Fh(t) and f0 = 1/T0.2-5 FOURIER SERIES2-5 FOURIER SERIESE

22、xample 2-12 SPECTRUM OF A PERIODIC RECTANGULAR WAVE2-5 FOURIER SERIES2-5 FOURIER SERIESFor a periodic waveform w(t), the normalized power is given bywhere the cn are the complex Fourier coefficients for the waveform.2-5 FOURIER SERIESFor a periodic waveform w(t), the PSD is given bywhere the cn are

23、the complex Fourier coefficients for the waveform.2-5 FOURIER SERIESExample 2-13 PSD OF A PERIODIC RECTANGULAR WAVE2-5 FOURIER SERIES2-6 REVIEW OF LINEAR SYSTEMSAn electronic filter or system is linear when superposition holds that is, when2-6 REVIEW OF LINEAR SYSTEMS2-6 REVIEW OF LINEAR SYSTEMSExam

24、ple 2-14 RC LOWPASS FILTER2-6 REVIEW OF LINEAR SYSTEMSExample 2-14 RC LOWPASS FILTER2-6 REVIEW OF LINEAR SYSTEMSExample 2-14 RC LOWPASS FILTER2-6 REVIEW OF LINEAR SYSTEMSA distortionless channel is often desired. This implies that the channel output is just proportional to a delayed version of the i

25、nputWhen the first condition is satisfied, there is no amplitude distortion. An LTI system will produce amplitude distortion if the amplitude response is not flat.When the second condition is satisfied, there is no phase distortion. An LTI system will produce phase distortion if the phase response i

26、s not a linear function of frequency.2-6 REVIEW OF LINEAR SYSTEMSExample 2-15 DISTORTION CAUSED BY AN RC LOWPASS FILTER2-6 REVIEW OF LINEAR SYSTEMSExample 2-15 DISTORTION CAUSED BY AN RC LOWPASS FILTER2-6 REVIEW OF LINEAR SYSTEMSExample 2-15 DISTORTION CAUSED BY AN RC LOWPASS FILTER2-6 REVIEW OF LIN

27、EAR SYSTEMSIn audio applications, the human ear is relatively sensitive to amplitude distortion, but insensitive to phase distortion. Thus, in linear distortion specifications of high-fidelity audio amplifiers, one is interested primarily in nonflat magnitude frequency response characteristics.In an

28、alog video applications, the opposite is true: The phase response becomes the dominant consideration.For data signals, a linear filter can cause a data pulse in one time slot to smear into adjacent time slots causing intersymbol interference (ISI).If the system is nonlinear or time varying, other ty

29、pes of distortion will be produced. As a result, there are new frequency components at the output that are not present at the input.2-7 BANDLIMITED SIGNALS AND NOISEA waveform w(t) is said to be (absolutely) bandlimited to B hertz ifA waveform w(t) is said to be (absolutely) time limited ifAn absolu

30、tely bandlimited waveform cannot be absolutely time limited, and vice versa. This raises an engineering paradox. This paradox is resolved by realizing that we are modeling a physical process with a mathematical model and perhaps the assumption in the model are not satisfied although we believe them

31、to be satisfied.If a waveform is absolutely bandlimited, it is analytic function.2-7 BANDLIMITED SIGNALS AND NOISE(Shannon) Sampling theorem. Any physical waveform can be represented over the interval - t 2B.T is selected to give the desired frequency resolution, where the frequency resolution is f

32、= 1/T.N is the number of data points and is dtermined by N = T/t.2-8 DISCRETE FOURIER TRANSFORMThe relationship between CFT and DFT is as follows:2-8 DISCRETE FOURIER TRANSFORM2-8 DISCRETE FOURIER TRANSFORMThe relationship between the coefficients for the complex Fourier series and DFT is as follows

33、:What is bandwidth? There are numerous definitions of the term.In engineering definitions, the bandwidth is taken to be the width of a positive frequency band. In other words, the bandwidth would be f2 f1, where f2 f1 0 and is determined by the particular definition that is used.ABSOLUTE BANDWIDTH i

34、s f2 f1 , where the spectrum is zero outside the interval f1 f f2 along the positive frequency axis.3-dB BANDWIDTH (or HALF-POWER BANDWIDTH) is f2 f1 , where for frequencies inside the band f1 f f2, the magnitude spectra, say, |H(f)|, falls no lower than 1/ times the maximum value of |H(f)| , and th

35、e maximum value occurs at a frequency inside the band.EQUIVALENT NOISE BANDWIDTH is the width of a fictitious rectangular spectrum such that the power in that rectangular band is equal to the power associated with the actual spectrum over positive frequencies. Let f0 be the frequency at which the ma

36、gnitude spectrum has a maximum, then2-9 BANDWIDTH OF SIGNALSNULL-TO-NULL BANDWIDTH (or ZERO-CROSSING BANDWIDTH) is f2 f1 , where f2 is the first null in the envelope of the magnitude spectrum above f0 and , for bandpass systems, f1 is the first null in the envelope below f0, where f0 is the frequenc

37、y where the magnitude spectrum is a maximum. For baseband systems f1 is usually zero.BOUNDED SPECTRUM BANDWIDTH is f2 f1 such that outside the band f1 f f2, the PSD must be down at least a certain amount, say, 50dB, below the maximum value of the PSD.POWER BANDWIDTH is f2 f1, where f1 f f2 defines t

38、he frequency band in which 99% of the total power resides. FCC BANWIDTH is an authorized bandwidth parameter assigned by the FCC to specify the spectrum allowed in communication systems. For operating frequencies below 15 GHz, in any 4kHz band, the center frequency of which is removed from the assigned frequency by P% (50 P 250) of the a