Chapter 2. Signals and Spectra

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1 Chapter 2 Signals and Spectra

2 Outline Properties of Signals and Noise Fourier Transform and Spectra Power Spectral Density and Autocorrelation Function Orthogonal Series Representation of Signals and Noise Fourier Series Linear Systems Bandlimited Signals and Noise Discrete Fourier Transform

3 Waveform Properties In communications, the received waveform basically comprises two parts: Desired signal or Information Undesired signal or Noise Assuming a signal is deterministic and physically realizable (measureable and contains only real part) Waveforms below to many different categories Physically realizable or non-physically realizable Deterministic or stochastic Analog or digital Power or energy Periodic or non-periodic

4 Waveform Characteristics (Definitions) p(t) = If w(t) is periodic with To, lim1/tà 1/To av Pnorm =Pav, when RLoad=1 norm Note: See Notes:2a W dc = w(t) = w(t) Note : w(t) can be v(t) or i(t) P av = p(t) = v(t)!i(t) Note: w 2 (t) = W rms

5 Real Meaning of RMS RMS for a set of n components RMS for continuous function from T1 to T2 RMS for a function over all the times

6 Energy & Power Waveforms Signal Definition: Energy _ Signal! 0 < E < " Power _ Signal! 0 < P < " Note that a signal can either have Finite total normalized energy or Finite average normalized power Note: If w(t) is periodic with To, lim1/tà 1/To

7 Example Voltage Periodic Signal!

8 Example (continued) Voltage * Instantaneous Power: Current Note: P av = p(t) = v(t)!i(t) Average Power Lamps flickering at 120 pulses/sec!

In this case V is the Peak amplitude of v(t) Current V rms = v 2 (t) = 1 T 0 V rms

9 Example (continued) Voltage * Instantaneous Power: Note that this is only true when V(t) is a sinusoidal. In this case V is the Peak amplitude of v(t) Current V rms = v 2 (t) = 1 T 0 V rms = V 2 ;I rms = I 2 ; To/2 "!To/2 [V cos(w o t)] 2 dt V = Vpeak Average Power P av = V rms # I rms = V # I 2 Lamps flickering at 120 pulses/sec!

10 Example - Scope Note that RMS is Vpeak/SQRT(2)

11 Example - Scope Note that RMS changes as the waveform Changes; independent of the frequency

12 Example - Scope Two signals being multiplied by each other!

13 Assume v(t) and i(t) are in phase. Plot the p(t). Example - Matlab

14 Example v(t)= e^t is a periodic voltage signal over time interval 0<t<1. Find DC & RMS values of the waveform

15 Decibel

16 Example Vrms=1.78V V(t) = e^t

17 Phasor Complex Number (y/x)

18 DEMO

19 Example - Scope Note that RMS is Vpeak/SQRT(2)

20 Example - Scope Note that RMS changes as the waveform Changes; independent of the frequency

21 Example - Scope Two signals being multiplied by each other!

22 Fourier Transform (1) How can ewe represent a waveform? Time domain Frequency domain à rate of occurrences Fourier Transform (FT) is a mechanism that can find the frequencies w(t): W(f) is the two-sided spectrum of w(t) à positive/neg. freq. W(f) is a complex function: Quadrature Components Phasor Components Time waveform can be obtained from spectrum using Inverse FT

23 Fourier Transform (2) Thus, Fourier Transfer Pair: w(t) ß à W(f) W(t) is Fourier transformable if it satisfies the Dirichlet conditions (sufficient conditions): Over a finite time interval w(t), is single valued with a finite number of Max & Min, & discontinuities. w(t) is absolutely integrable: * dir-i-kley

24 Dirac Delta and Unit Step Functions u(t) Note that Shifting Property of Delta Function * dih-rak

25 FT Examples (1) Note that in general: In our case, to = 0 and f(to) =1 See Appendix A of the Textbook! NEXTà

26 FT Example (2) Magnitude-Phase Form: Note: Pay attention to how the equations are setup!

27 Phase Difference & Time Delay What does time delay have to do with phase angle?

28 Spectral symmetry of real signals: If w(t) is real, w(t) = w*(t) then Properties of FT Svcv If w1(t)=w2(t)=w(t) à dfdf à

29 Other FT Properties Find FT of w(t)sin(w c t)! w(t)sin(w c t) = wt(cos(wct-90) = ½ [-j W(f-fc) + j W(f+fc]

30 Spectrum of A Sinusoid Given v(t) = Asin(w o t) the following function plot the magnitude spectrum and phase Spectrum of v(t): v(f) & θ(f) v(t) Note that V(f) is purely imaginary à When f>0, then θ(f)= -π/2 à When f<0, then θ(f)= +π/2 Similar to FT for DC waveform Example

31 Other Fourier Transform Pairs (1) Note: sinc (0) à 1 & Max value of sinc(x)à 1/x Sa stands for Sampling Function

32 Other Fourier Transform Pairs (2) Using Duality Property Note:

33 Other Fourier Transform Pairs (3)

34 Examples Using time delay property For 8sin(6πt), we have: Note: 2πfo=2π(3) For what freq. W(f) has its max?

35 Plotting Magnitude and Phase Spectrum

36 Spectrum of Rectangular Pulses What is w(t)?

37 Spectrum of Rectangular Pulses

38 Spectrum of a Switched Sinusoid Waveform of a switch sinusoid can be represented as follow: The frequency domain representation of w(t) will be: Note that the spectrum of w(t) is imaginary! As Tà INF, 1/Tà 0, then Sa waveform converges to a doublesided delta waveform Magnitude Spectrum of w(t)

Alternative Tools Try the following: arg(1/(1+ix)); -100<x<100 magnitude(1/(1+ix)); http://www.wolframalpha.com/input/?

39 Alternative Tools Try the following: arg(1/(1+ix)); -100<x<100 magnitude(1/(1+ix)); Another very interesting tool to demonstrate FT: Try the following: - sin(10*x)+sin(100*x) - sin(10*x)+sin(100*x) - exp(0.05*x)*sin(100*x)

Power Spectral Density How the power content of signals and noise is distributed over different frequencies Useful in describing how the power content of signal

spectrum of w(t) due to absolute value operation If the PSD is plotted in db units, the plot of the PSD is identical to the plot of the Magnitude Spectrum in db

40 Power Spectral Density How the power content of signals and noise is distributed over different frequencies Useful in describing how the power content of signal with noise is affected by filters & other devices Important properties: PSD is always a real nonnegative function of frequency PSD is not sensitive to the phase spectrum of w(t) due to absolute value operation If the PSD is plotted in db units, the plot of the PSD is identical to the plot of the Magnitude Spectrum in db units PSD has the unit of watts/hz (or, equivalently, V 2 /Hz or A 2 /Hz) Direct Method! W T (t) is the truncated version of the signal: Any other way we can find PSD?à

41 Autocorrelation Function Direct Method! Indirect Method! avg PSD

42 Example: Power Spectrum of a Sinusoid Find the PSD of Method 2: using the indirect method (finding the autocorrelation): We can verify this by

43 Orthogonal & Orthonormal Functions Over some interval a & b Orthogonal functions are independent, in disagreement, unlikely! & Kn = 1 Note that if Kn is any constant other than unity, then the functions are not orthonormal!

44 Example Seems like two functions are always orthogonal!!!! Can you show this?

45 Orthogonal Series Note that a n and φ n (t) are orthogonal w(t) à orthogonal series à

46 Example Are sets of complex exponential functions ( ) over the interval a<t<b=a+to, w o =2π/To orthogonal? Are they orthonormal?

47 Fourier Series (Avg Pwr)

48 FS for Periodic Functions We can represent all periodic signals as harmonic series of the form C n are the Fourier Series Coefficients & n is real n=0 à Cn=o which is the DC signal n=+/-1 yields the fundamental frequency or the first harmonic ω 0 n >=2 harmonics FOR PERIODIC SINUSOIDAL SIGNALS:

49 Fourier Series and Frequency Spectra We can plot the frequency spectrum or line spectrum of a signal In Fourier Series n represent harmonics Frequency spectrum is a graph that shows the amplitudes and/or phases of the Fourier Series coefficients Cn. Phase spectrum φn The lines Cn are called line spectra because we indicate the values by lines Cn n

50 Different Forms of Fourier Series Fourier Series representation has different forms: Note that n=k Polar Form Quadrature Form What is the relationship between them?à Finding the coefficients!

51 Fourier Series in Quadrature & Polar Forms Also Known as Trigonometric Form Slightly different notations! Note that n=k Also Known as Combined Trigonometric Form

52 Euler s Relationship Review Euler formulas Important Relationships

53 Find Fourier Series Coefficients for Examples of FS Find Fourier Series Coefficients for Remember:

54 Example Given the following periodic square wave, find the Fourier Series representations and plot Ck as a function of k. (Rectangular waveform) 1 C k = T T o sinc Tk! o 2! x(t) = = 2T 1 T o sinc(t 1 k! o ) # 2T $ 1 sinc(t 1 k! o ) e j! otk T o k="# Sinc Function X(f) Note that: T 1 = T/4 = T o /4 w o =2π/T = 2π/T o Note: sinc (infinity) à 1 & Max value of sinc(x)à 1/x

55 Example Find the Fourier coefficients for the periodic rectangular wave shown here: C n = Magnitude Spectrum

56 PSD Of A Periodic Square Waveform Make sure you know the difference between Frequency Spectrum, Magnitude Frequency Spectrum, and Power Spectral Density

57 Same Example A different Approach Note that here we are using quadrature form of shifted version of v(t): What is the difference? Note that N=n; T=To

58 A Closer Look at the Quadrature Form of FS Consider the following quadrature FS representation of an odd square waveform with no offset: 4V/π 4V/3π Thus: Cn = 4V/Nπ 4V/5π 4V/7π Magnitude Frequency Spectrum W(f)=FS[v(t)] N represents the Harmonic NUMBER N=1 freq=f=1/t 3 3f 5 5f 7 7f Nà fà How does the time-domain waveform of the first 7 harmonics look like?

59 Generating an Square Wave N=1,3,5 N=1,3,5, 7 This is how the time-domain waveform of the first 7 harmonics looks like! Frequency Components of Square Wave N=1,3,5, 7, 9,.. Fourier Expansion

60 What Is the FS of A Pulse Signal? Note that the width of the pulse can change! 0.25=τ/Τ 10=1/τ 20=2/τ 30=3/τ Magnitude Line Spectra of the pulse signal note that the envelope is a sinc ( ) function! What happens to the envelope as the pulse gets smaller?

61 Bandlimiting Effects on Signals All communication systems have some finite bandwidth Sufficient BW must be guaranteed to reserve the signal integrity 1 KHz Square Wave 1 KHz Square Wave band-limited to 8KHz 1 st, 3 rd, & 5 th Harmonics 1 KHz Square Wave band-limited to 6KHz First & 3rd Harmonics 1 KHz Square Wave band-limited to 4KHz First Harmonic 1 KHz Square Wave band-limited to 2KHz

62 Bandlimiting in Mixing Devices Mixing is the process of combining two or more signals (e.g., Op-Amps) Linear Summing Amplifiers with single inputs Amplifiers with multiple inputs Nonlinear Summing Amplifiers with single inputs Amplifiers with multiple inputs Amplified by A

63 Bandlimiting in Mixing Devices Mixing is the process of combining two or more signals (e.g., Op-Amps) Linear Summing Amplifiers with single inputs Amplifiers with multiple inputs Nonlinear Summing Amplifiers with single inputs Amplifiers with multiple inputs For nonlinear case an infinite number of harmonic frequencies are produced! If these cross-products are undesired à we call them intermodulation distortion! v out = Av in + Bv in 2 + Cv in 3 v in = V a sin(2! f a t)+v b sin(2! f b t)!! > v out = A(V a sin(2! f a t)+v b sin(2! f b t)) +B(V a sin(2! f a t)+v b sin(2! f b t)) 2 +C(V a sin(2! f a t)+v b sin(2! f b t)) 3 If these cross-products are desired à we call them modulation! Cross-Products Cross-Products = mfa +/- nfb

64 Example Assume we have a nonlinear system receiving two tones with frequencies of 5KHz and 7 KHz. Plot the output frequency spectrum for the first three harmonics (assume m & n can each be 1 & 2). Cross-Products = mfa +/- nfb n=1 & m=1 à 5+/-7=12KHz & 2KHz n=1 & m=2 à 5+/-14=9KHz & 19KHz n=2 & m=1 à 10+/-7=3KHz & 17KHz n=2 & m=2 à 10+/-14=24KHz & 4KHz Fundamental frequencies (first harmonic): 5KHz & 7KHz Harmonics: Second harmonic: 10KHz & 14KHz Third harmonic: 15KHz & 21KHz All together there are 14 frequencies on the frequency spectrum!

65 References Leon W. Couch II, Digital and Analog Communication Systems, 8 th edition, Pearson / Prentice, Chapter 1

Chapter 2. Fourier Series & Fourier Transform. Updated:2/11/15

Chapter 2. Fourier Series & Fourier Transform. Updated:2/11/15 Chapter 2 Fourier Series & Fourier Transform Updated:2/11/15 Outline Systems and frequency domain representation Fourier Series and different representation of FS Fourier Transform and Spectra Power Spectral