Introduction to Mathematical Modeling of Signals and Systems Mathematical Representation of Signals Signals represent or encode information In communications applications the information is almost always encoded In the probing of medical and other physical systems, where signals occur naturally, the information is not purposefully encoded In human speech we create a waveform as a function of time when we force air across our vocal cords and through our vocal tract!"#$%&'()'*+"),- %'*.+&/+0"/)+ -'*0"(&+--&+ 2&'#"/)+".'%,3 /&,%/"$*/'",* +3+%/&$%,3"-$4*,3 /),/".,&$+-"'.+& /$#+5"!
Introduction to Mathematical Modeling of Signals and Systems Signals, such as the above speech signal, are continuous functions of time, and denoted as a continuous-time signal The independent variable in this case is time, t, but could be another variable of interest, e.g., position, depth, temperature, pressure The mathematical notation for the speech signal recorded by the microphone might be %" In order to process this signal by computer means, we may sample this signal at regular interval, resulting in %' %'& % (.2) The signal %' is known as a discrete-time signal, and & % is the sampling period Note that the independent variable of the sampled signal is the integer sequence '! & " &! Discrete-time signals can only be evaluated at integer values & % -2
The speech waveform is an example of a one-dimensional signal, but we may have more that one dimension An image, say a photograph, is an example of a two-dimensional signal, being a function of two spatial variables, e.g. (!) If the image is put into motion, as in a movie or video, we now have a three-dimensional image, where the third independent variable is time,!)" Note: movies and videos are shot in frames, so actually time is discretized, e.g., " '& % (often & & % '" fps) To manipulate an image on a computer we need to sample the image, and create a two-dimensional discrete-time signal (*' (*! ' ) (.3) where m and n takes on integer values, and! and ) represent the horizontal and vertical sampling periods respectively Mathematical Representation of Systems In mathematical modeling terms a system is a function that transforms or maps the input signal/sequence, to a new output signal/sequence )" & +!" )' &,!' (.4) where the subscripts c and d denote continuous and discrete system operators 3
Introduction to Mathematical Modeling of Signals and Systems Because we are at present viewing the system as a pure mathematical model, the notion of a system seems abstract and distant Consider the microphone as a system which converts sound pressure from the vocal tract into an electrical signal Once the speech waveform is in an electrical waveform format, we might want to form the square of the signal as a first step in finding the energy of the signal, i.e., )"!"! (.5)!"-6,&+&"-7-/+# )"!"! The squarer system also exists for discrete-time signals, and in fact is easier to implement, since all we need to do is multiply each signal sample by itself 4
)' (.6) If we send )' through a second system known as a digital filter, we can form an estimate of the signal energy This is a future topic for this course Thinking About Systems!'!!'!' Engineers like to use block diagrams to visualize systems Low level systems are often interconnected to form larger systems or subsystems Consider the squaring system!" & ((( )" T"$-","4+*+&$%"-7-/+#!" (((! )" The ideal sampling operation, described earlier as a means to convert a continuous-time signal to a discrete-times signal is represented in block diagram form as an ideal C-to-D converter!" 80+,3 9:/':; 9'*.+&/+& & %!'!'& %!"-7-/+#"(,&,#+/+&"/),/ -(+%$2$+-"/)+"-,#(3+"-(,%$*4 5
Introduction to Mathematical Modeling of Signals and Systems A more complex system, depicted as a collection of subsystem blocks, is a system that records and then plays back an audio source using a compact disk (CD) storage medium The optical disk reader shown above is actually a high-level block, as it is composed of many lower-level subsystems, e.g., Laser, on a sliding carriage, to illuminate the CD An optical detector on the same sliding carriage A servo control system positions the carriage to follow the track over the disk A servo speed control to maintain a constant linear velocity as /0 data is read from different portions of the disk more... The Next Step Basic signals, composed of linear combinations of trigonometric functions of time will be studied next We also consider complex number representations as a means to simplify the combining of more than one sinusoidal signal 6
Sinusoids A general class of signals used for modeling the interaction of signals in systems, are based on the trigonometric functions sine and cosine The general mathematical form of a single sinusoidal signal is xt Acos 0 t + (2.) where A denotes the amplitude, 0 is the frequency in radians/s (radian frequency), and is the phase in radians The arguments of cos and sin are in radians We will spend considerable time working with sinusoidal signals, and hopefully the various modeling applications presented in this course will make their usefulness clear Example: xt 0cos2440t 0.4 The pattern repeats every 440 0.00227 2.27ms This time interval is known as the period of xt 2
Review of Sine and Cosine Functions The text discusses how a tuning fork, used in tuning musical instruments, produces a sound wave that closely resembles a single sinusoid signal In particular the pitch A above middle C has an oscillation frequency of 440 hertz Review of Sine and Cosine Functions Trigonometric functions were first encountered in your K 2 math courses The typical scenario to explain sine and cosine functions is depicted below 2 2 The right-triangle formed in the first quadrant has sides of length x and y, and hypotenuse of length r The angle has cosine defined as x/r and sine defined as y/ r The above graphic also shows how a point of distance r and angle in the first quadrant of the x y plane is related
Review of Sine and Cosine Functions to the x and y coordinates of the point via sin( ) and cos( ), e.g., xy rcos rsin (2.2) Moving beyond the definitions and geometry interpretations, we now consider the signal/waveform properties The function plots are identical in shape, with the sine plot shifted to the right relative to the cosine plot by 2 This is expected since a well known trig identity states that sin cos 2 (2.3) We also observe that both waveforms repeat every 2 radians; read period 2 Additionally the amplitude of each ranges from - and A few key function properties and trigonometric identities 2 3
Review of Sine and Cosine Functions are given in the following tables Property Table 2.: Some sine and cosine properties Equation Equivalence sin cos 2 or cos sin+ 2 Periodicity cos 2k cos, when k is an integer; holds for sine also Evenness of cosine cos cos Oddness of sine sin sin Number 2 3 4 5 6 7 Table 2.2: Some trigonometric identities sin 2 Equation + cos 2 cos2 cos 2 sin 2 sin2 2sincos sin sincoscossin cos coscossinsin cos 2 sin 2 -- + cos2 2 For more properties consult a math handbook -- cos2 2 2 4
Review of Sine and Cosine Functions The relationship between sine and cosine show up in calculus too, in particular dsin dcos ------------- cos and -------------- d d sin (2.4) This says that the slope at any point on the sine curve is the cosine, and the slope at any point on the cosine curve is the negative of the sine Example: Prove Identity #6 Using Identities # and #2 If we add the left side of to the right side of 2 we get 2cos 2 + cos2 or cos 2 -- + cos2 2 (2.5) Example: Find an expression for cos8 in terms of cos9, cos7, and cos using #5 Let 8 and, then write out #5 under both sign choices cos8+ cos8cos sin8sin + cos8 cos8cos + sin8sin cos9 + cos7 2cos8cos or cos9 + cos7 cos8 ------------------------------------ 2cos (2.6) (2.7) 2 5
Review of Complex Numbers Review of Complex Numbers See Appendix A of the text for more information A complex number is an ordered pair of real numbers denoted z x y The first number, x, is called the real part, while the second number, y, is called the imaginary part For algebraic manipulation purposes we write xy x+ iy x + jy where i j ; electrical engineers typically use j since i is often used to denote current Note: j j The rectangular form of a complex number is as defined above, The corresponding polar form is z x y x + jy z re j r ze jargz.tom M. Apostle, Mathematical Analysis, second edition, Addison Wesley, p. 5, 974. 2 6
We can plot a complex number as a vector Review of Complex Numbers xy Example:: z 2 + j5, z 4 j3, z 5 + j0, z 3 j3 2 7
Review of Complex Numbers Example:: z 245, z 350, & z 3 80 For complex numbers z x + jy and z 2 x 2 + jy 2 we define/calculate z + z 2 x + x 2 + jy + y 2 (sum) z z 2 x x 2 + jy y 2 (difference) z z 2 x x 2 y y 2 + jx y 2 + y x 2 (product) z ---- z 2 x x 2 + y y 2 jx y 2 y x 2 ------------------------------------------------------------------------ (quotient) x2 2 + y2 2 ECE 260 Signals and Systems 2 8
Review of Complex Numbers 2 z x 2 + y (magnitude) z tan y x (angle) z* x jy (complex conjugate) MATLAB is also consistent with all of the above, starting with the fact that i and j are predefined to be rectangular polar To convert from polar to rectangular we can use simple trigonometry to show that x rcos y rsin (2.24) Similarly we can show that rectangular to polar conversion is r x 2 + y 2 tan y x note add outside Q & Q4 (2.25) 2 9
Review of Complex Numbers Example: Rect to Polar and Polar to Rect Consider z 2 + j5 In MATLAB we simply enter the numbers directly and then need to use the functions abs() and angle() to convert >> z 2 + j*5 z 2.0000e+00 + 5.0000e+00i >> [abs(z) angle(z)] ans 5.3852e+00 Using say a TI-89 calculator is similar.903e+00 % mag & phase in rad Consider z 2 245 In MATLAB we simply enter the numbers directly as a complex exponential >> z2 2*exp(j*45*pi/80) z2.442e+00 +.442e+00i 2 0
Review of Complex Numbers Using the TI-89 we can directly enter the polar form using the angle notation or using a complex exponential Example: Complex Arithmetic Consider z + j7 and z 2 4 j9 Find z + z 2 >> z +j*7; >> z2-4-j*9; >> z+z2 ans -3.0000e+00-2.0000e+00i Using the TI-89 we obtain 2
Review of Complex Numbers Find z z 2 >> z*z2 ans 5.9000e+0-3.7000e+0i Using the TI-89 we obtain Find z z 2 >> z/z2 ans -6.9072e-0 -.9588e-0i TI-89 Results Euler s Formula: A special mathematical result, of special importance to electrical engineers, is the fact that e j cos + jsin (2.26) 2 2
Turning (2.26) around yields (inverse Euler formulas) sin e j It also follows that e j Sinusoidal Signals Sinusoidal Signals --------------------- and cos --------------------- + (2.27) 2j 2 e j z x+ jy rcos + jrsin e j (2.28) A general sinusoidal function of time is written as xt Acos 0 t + Acos2f 0 t + (2.29) where in the second form 0 2f 0 Since cos it follows that xt swings between A, so the amplitude of xt is A The phase shift in radians is, so if we are given a sine signal (instead of the cosine version), we see via the equivalence property that xt Asin 0 t + Acos 0 t + 2 (2.30) which implies that 2 Engineers often prefer the second form of (2.8) where the oscillation frequency in cycles/s. 0 ----- rad/s ---------- 2 rad f sec 0 f 0 is 2 3
Sinusoidal Signals Example: xt 20cos240t 0.4 Clearly, A 20, f 0 40 cycles/s, and 0.4 rad Maxima Interval (period) ----- 40 0.025s 25ms Since this signal is periodic, the time interval between maxima, minima, and zero crossings, for example, are identical Relation of Frequency to Period A signal is periodic if we can write xt + T 0 xt (2.3) where the smallest T 0 satisfying (2.0) is the period For a single sinusoid we can relate T 0 to f 0 by considering xt + T 0 xt Acos 0 t + T 0 + Acos 0 t + cos 0 t + + 0 T 0 cos 0 t + (2.32) From the periodicity property of cosine, equality is maintained if cos2k cos, so we need to have 2 4
Sinusoidal Signals 0 T 0 2 T 0 or 2f 0 T 0 T 0 2 ----- 0 --- f 0 (2.33) So we see that T 0 and f 0 are reciprocals, with the units of T 0 being time and the units of f 0 inverse time or cycles per second, as stated earlier In honor of Heinrich Hertz, who first demonstrated the existence of radio waves, cycles per second is replaced with Hertz (Hz) 2 5
Sinusoidal Signals Example: 5cos2f 0 t with f 0 200, 00, and 0 Hz Period doubles as frequency halves A constant signal as the oscillation frequency is zero The inverse relationship between time and frequency will be explored through out this course 2 6
Phase Shift and Time Shift Sinusoidal Signals We know that the phase shift parameter in the sinusoid moves the waveform left or right on the time axis To formally understand why this is, we will first form an understanding of time-shifting in general Consider a triangularly shaped signal having piece wise continuous definition st 2t, 0 t 2 -- 4 2t,2 t 2 3 0, otherwise (2.34) 2t st -- 4 2t 3-2 0 -- 2 3 t Now we wish to consider the signal x t st 2 As a starting point we note that st is active over just the interval 0 t 2, so with t t 2 we have 0 t 2 2 2 t 4 (2.35) which means that x t is active over 2 t 4 The piece wise definition of x t can be obtained by direct substitution of t 2 everywhere t appears in (2.34) 2 7
Sinusoidal Signals x t x t st 2 2t 2 2t 2, 0t 2 2 -- 4 2t 2,2 t 2 2 3 0, otherwise 2t 4, 2 t 5 2 -- 8 2t,52 t 4 3 0, otherwise -- 8 2t 3 (2.36) 0 2 5 3 4 -- 2 t In summary we see that the original signal the right by 2 s st is moved to Example: Plot With t t+ by one second st + st + we expect that the signal will shift to the left - -- 2 0 2 3 t 2 8
Sinusoidal Signals The new equations are obtained as before 0 t + 2 t so (2.37) st + 2t +, 0 t + 2 -- 4 2t +,2 t + 2 3 0, otherwise 2t + 2, t 2 -- 2 2t, 2 t 3 0, otherwise (2.38) Modeling time shifted signals shows up frequently In general terms we say that x t st t (2.39) is delayed in time relative to st if t 0, and advanced in time relative to st if t 0 A cosine signal has positive peak located at If this signal is delayed by the peak shifts to the right and the corresponding phase shift is negative Consider t x 0 t Acos 0 t t 0 2 9
Sinusoidal Signals x 0 t t Acos 0 t t Acos 0 t 0 t (2.40) which implies that in terms of phase shift we have 0 t For a given phase shift we can turn the above analysis around and solve for the time delay via t ----- 0 ---------- 2f 0 (2.4) Since T 0 f 0, we can also write the phase shift in terms of the period 2f 0 t 2 t ----- T 0 (2.42) An important point to note here is that both cosine and sine are mod 2 functions, meaning that phase is only unique on a 2 interval, say ( ] or (0 2] Example: Suppose t 0ms and T 0 3ms Direct substitution into (2.2) results in 2 0 ----- 20 ----- 6.6667 3 3 (2.43) We need to reduce this value modulo 2 to the interval ( ] by adding (or subtracting as needed) multiples of 2 The result is the reduced phase value 2 20
Sampling and Plotting Sinusoids 20 20 + 8 2 ----- + 6 ----------------- -- 0.6667 3 3 3 Does this result make sense? (2.44) A time delay of 0 ms with a period of 3 ms means that we have delayed the sinusoid three full periods plus ms A ms delay is /3 of a period, with half of a period corresponding to rad, so a delay of /3 period is a phase shift of 2 3 0.6667; agrees with the above analysis Modulo the period delay of ms Actual Delay of 0 ms t (ms) Blue no delay Red 0 ms Delay The value of phase shift that lies on the interval known as the principle value Sampling and Plotting Sinusoids is When plotting sinusoidal signals using computer tools, we are also faced with the fact that only a discrete-time version 2 2
Sampling and Plotting Sinusoids of xt Acos2f 0 t + may be generated and plotted This fact holds true whether we are using MATLAB, C, Mathematica, Excel, or any other computational tool When t nt s we need to realize that sample spacing needs to be small enough relative to the frequency f 0 such that when plotted by connecting the dots (linear interpolation), the waveform picture is not too distorted In Chapter 4 we will discuss sampling theory, which will tell us the maximum sample spacing (minimum sampling rate which is T s ), such that the sequence xn xnt s can be used to perfectly reconstruct xt from xn For now we are more concerned with having a good plot appearance relative to the expected sinusoidal shape A reasonable plot can be created with about 0 samples per period, that is with T s 0f 0 T 0 0 We will now consider several MATLAB example plots >> t 0:/(5*3):; x 5*cos(2*pi*3*t-.5*pi); >> subplot(3) >> plot(t,x,'.-'); grid >> xlabel('time in seconds') >> ylabel('amplitude') >> t 0:/(0*3):; x 5*cos(2*pi*3*t-.5*pi); >> subplot(32) 2 22
Sampling and Plotting Sinusoids >> plot(t,x,'.-'); grid >> xlabel('time in seconds') >> ylabel('amplitude') >> t 0:/(50*3):; x 5*cos(2*pi*3*t-.5*pi); >> subplot(33) >> plot(t,x,'.-'); grid >> xlabel('time in seconds') >> ylabel('amplitude') >> print -depsc -tiff sampled_cosine.eps Amplitude Amplitude Amplitude 20 0 0 0 20 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time in seconds 20 0 0 0 20 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time in seconds 20 0 0 5 Samples per period 0 Samples per period f 0 3 Hz, A 5, -/2 T T 0 s ----- 5 T T 0 s -------- 0 T T 0 s ----- 50 0 50 Samples per period 20 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time in seconds 2 23
Complex Exponentials and Phasors!"#$%&'()'$"*&*+,-%.(-*/(0-."2. Modeling signals as pure sinusoids is not that common. We typically have more that one sinusoid present. Manipulating multiple sinusoids is actually easier when we form a complex exponential representation. Complex Exponential Signals Motivated by Euler s formula above, and the earlier definition of a cosine signal, we define the complex exponential signal as where!" #$ %!" +!" # and!" "#$!"! " + Note that using Euler s formula!" #$ %!" + #%&'! " + + %#'()! " + (2.44) (2.45) We see that the complex sinusoid has amplitude A, phase shift, and frequency! rad/s Note in particular that *+!" #%&'! " +,-!" #'()! " + (2.46) 3:3;
!"#$%&'()'$"*&*+,-%.(-*/(0-."2. The result of (2.46) is what ultimately motivates us to consider the complex exponential signal We can always write &" *+ #$ %!" + #%&'! " + (2.47) The Rotating Phasor Interpretation Complex numbers in polar form can be easily multiplied as!. ' / $ % / ' 0 $ % 0 ' / ' 0 $ % / 0! /! 0 + (2.48) For the case of!" #$ %!" + (2.49) 3:3<
we can write where!" #$ % $ %!" ($ %!" ( #$ %!"#$%&'()'$"*&*+,-%.(-*/(0-."2. (2.50) The complex amplitude ( is called the phasor, as it is the gain and phase value applied to the time varying component to form!" $ %!" This is common terminology is electrical engineering circuit theory The time varying term has unit magnitude and rotates counter clockwise in the complex plane at a rate of! rad/s ( rotations/s) )! $ %!" The time duration for one rotation is the period *! / )! The combination (product) of the fixed phasor ( and $ %!" results in a rotating phasor For positive frequency! the rotation is counter clockwise, and for negative frequency the rotation is clockwise #$ %&'()(*" +,"-."/0 2"34)(*" +,"-."/0 " "!"!"!"#$#%&'()*$+",+ "! " + 3:34
!"#$%&'()'$"*&*+,-%.(-*/(0-."2. Example:!" +230" Plot a series of snap shots of the rotating phasor when * + / 8 (note *! / s)!""""""""""""""""""""""""""""""""""""""""""""""""""""""!#$#%&'()*#+(,-#+.'#/-0-'*(0/##%-23-0&-#.+#'.**(0/#!#)4%.'#%0)#%4.*%!!!"""""""""""""""""""""""""""""""""""""""""""""""""""""" #!#/-*#*4-#+.&3%#.+#+(/3'-#<(0.<#>?#.'#&'-*-#!#(+#0.*#&'-*-@ +(/3'-A?B# &,+A?BC#!#&,-'#+(/3'-#<(0.<#>? # $#D#?@:C#+:#D#?C#)4(#D#E)("FC G#D#HC#!#&'-*-#H#I-&*.'#),.*% J%#D#?"HC +.'#0#D#:KGE? ####%3L),.*AF8980M?B ####*#D#:K?"9::K?C ####),.*A&.%A9N)(N*B8%(0A9N)(N*B8O6KOB ####4.,#.0 ####P#D#$N-Q)ARNA9N)(N+:N0NJ%M)4(BBC ####),.*AS:8'-,APBT8S:8(U/APBT8OV(0-7(*4O8?B ####!#(0%(-#%)'(0*+#&'-*-%##+.'U**-#%*'(0/ ####*(*,-A%)'(0*+AOJ(U-#D#!?@F+#%O80NJ%BBC ####Q(%A?@?NSE$#$#E$#$TBC#Q(%#-23,C ####),.*A'-,APB8(U/APB8O'@O8O5'6-'W(P-O8?HB ####4.,#.++ -0 3:3
!"#$%&'()'$"*&*+,-%.(-*/(0-."2. Time 0.0000 s Time 0.250 s 0 2 0 2 0 $ % $ %! 2 0 2 0 Time 0.2500 s $ % 0 Time 0.3750 s 2 0 2 2 0 2 Time 0.5000 s Time 0.6250 s 0 0 2 0 2 2 0 2 Time 0.7500 s Time 0.8750 s 0 0 2 0 2 2 0 2 3:3>
0-."2(?//,+,"* The inverse Euler formulas can be used to see that a cosine signal is composed of positive and negative frequency exponentials #%&'! " + # $%!" + $ %!" + + ---------------------------------------------------- 0 Phasor Addition / --($ %!" / --( 4 $ %!" + 0 0 / --! " 0 + *+!" / --! 4 " 0 (2.5) We often have to deal with multiple sinusoids. When the sinusoids are at the same frequency, we can derive a formula of the form - # %&',! " +, #%&'! " +, / (2.52) At present we have only the trig identities to aid us, and this approach becomes very messy for large N. Phasor Addition Rule We know that when complex numbers are added we must add real and imaginary parts separately Consider the sum 3:3@
0-."2(?//,+,"* (2.53) The above is valid since the real and imaginary parts add independently, that is and the same holds for the imaginary part (2.54) Secondly, a real sinusoid can always be written in terms of a complex sinusoid via Proof: - - #, $ j, (, (, / - *+ (,, / -, / -, / #$ % *+ (, #%&'! " + *+ #$ %!" + # %&',! " +, *+ #, $ %!",, / -, / + (2.55) +&55&6'78,&$79:;<4> - *+ #, $ % %,! " $, / *+ #$ % $ %!" *+ #$ %!" + #%&'! " + 3:A6
0-."2(?//,+,"* Example: Phasor Addition Rule in Action Consider the sum &" & / " + & 0 " 56%&'.!" +.6 /7! 8 + 950 %&'.!" + 7! /7! The frequency of the sinusoids is 5 Hz Using phasor notation we can write that & / " *+ 56$ & 0 " *+ 950$ so in the phasor addition rule %.6 /7! %.6 /7! %7! /7! ( / 56$ ( 0 950$ $ %.!" $ %.!" %7! /7! (2.56) (2.57) (2.58) We perform the complex addition and conversion back to polar form using the TI-89 ( 8()8#":(")' so ( ( / + ( 0 5;.<6 + %;5<9/9/ %<05;<!0 /7! /!57679$ (2.59) 3:A5
Finally, 0-."2(?//,+,"* (2.60) We can check this by directly plotting the waveform in MAT- LAB &" /!57<%&'.!" + <05;<!0 /7! XX#*#D#:K?"AY:N?YBK:@9C XX#Q?#D#F@YN&.%AZ:N)(N*MZYN)("?H:BC XX#Q9#D#[@9N&.%AZ:N)(N*MH:N)("?H:BC XX#Q#D#Q?MQ9C#!),.*#3%(0/#4.,#0#,(0-#%*\,-% 0?@;A:: &" x (t) x 2 (t) x(t) 5 Amplitude 0 & / " B??;CD7$' 5 & 0 " 0 0.06 0.04 0.02 0 0.02 0.04 0.06 Time in seconds The measured amplitude, 0.822, is close to the expected value 3:A3
The location of the peak can be converted to phase via 0-."2(?//,+,"* 0 --------------- //5<9 /5/!8#": <.5!0 <<5<9 (2.6) Summary of Phasor Addition When we need to form the sum of sinusoids at the same frequency, we obtain the final amplitude A and phase via ( ( / + ( 0 + + ( - #$ % (2.62) where (, #, $ %, and - &" #, %&'! " +,, / #%&'! " + (2.63) Example: &" *+.$ % 0) " + --! 0 % 0) " --! + 6$ +. + %0$ %0)!" Find ( ( / + ( 0 + (. From the given &" % -- 0 ( /.$ we observe that % -- ( 0 6$ (.. + %0 3:AA
09.,B.("C(+&(DE*,*8(F"2G To perform the complex addition we will work step-by-step To add complex numbers we convert to rectangular form Now, ( /.%&'-- + %.'()-- %. 0 0 ( 0 6%&' -- + %6'() --.56.66 %.56.66 (.. + %0 ( %. +.56.66 %.56.66 +. + %0 <56.66 + %/5<6 For use in the phasor sum formula we likely need the answer in polar form ( <56.66 0 + /5<6 0 "")--------------- /5<6 <56.66 <5<;9<!500! <5<;9<$ %!500! Physics of the Tuning Fork The tuning fork signal generation example discussed earlier was important because it is an example of a physical system that when struck, produces nearly a pure sinusoidal signal. By pure we mean a signal composed of a single frequency sinusoid, no other sinusoids at other frequencies, say harmonics (multiples of ) are present )! 3:A;
Equations from Laws of Physicss A 2-D model of the tuning fork is shown below E6&7)(/"'7&8 )F"7)./(/378&,G 09.,B.("C(+&(DE*,*8(F"2G When struck the vibration of the metal tine moves air molecules to produce a sound wave Hooke s law from physics (springs, etc.) says that the force to restore the tine back to its original &! position is the same as the original deformation (striking force), except for a sign change,.,& where k is the material stiffness constant (2.64) The acceleration produced by the restoring force (Newton s second law) is 3:A<
09.,B.("C(+&(DE*,*8(F"2G. /0 / 0 & ------- " 0 To balance the two forces (sum is zero), we must have / 0 ------- & " 0,& " (2.65) (2.66) General Solution to the Differential Equation To solve this equation we can actually guess the solution by inserting a test function of the form &" %&'! " 0 -------------- &" " 0 ---- "! '()! " 0! %&'! " We now plug this result into (2.66) to obtain / 0 ------- & " 0,& " 0 /! %&'! ",%&'! " which tells us that we must have /! 0, (2.67) (2.68) so it must be that,! --- / (2.69) 3:A4
09.,B.("C(+&(DE*,*8(F"2G This tells us that the oscillation frequency of the tuning fork is related to the ratio of the stiffness constant to the mass Greater stiffness means a higher oscillation frequency Greater mass means a lower oscillation frequency In terms of a real sinusoid the sound wave, to within a phase shift constant is of the form &" #%&' ---", + / (2.70) The sound produced by the 440Hz tuning fork was captured using MATLAB on a PC with a sound card and microphone The results were converted to double precision and saved in a.mat file along with a time axis vector Amplitude Amplitude 0.5 0 0.5 XX#,.#*30(0/+.'6 XX#),.*A*8QB 0 2 3 4 5 6 7 8 9 0 Time in seconds H&&$7IJKJ;@:L7' 0.5 0 0.5 4 4.002 4.004 4.006 4.008 4.0 4.02 4.04 4.06 4.08 4.02 Time in seconds 3:A
09.,B.("C(+&(DE*,*8(F"2G How pure is the signal produced by the tuning fork? In Chapter 3 of the text we begin a study of spectrum representation The zoom of the captured signal looks like a single sinusoid, but spectral analysis can be more revealing Consider the use of MATLAB s power spectral density function )%AQ8G++*8+%U)B (Detail comes later Chapter) XX#)%AQ89]?98H:::B 40 JJ@7MN78./O4$"/)457)./(/378&,G7P()0F Power Spectrum Magnitude (db) 20 0 20 40 AA@7MN7'"0&/O7F4,$&/(0 Q)F",7F4,$&/(0' 60 R."7)&7')4,)B.P7),4/'("/) 80 0 500 000 500 2000 2500 3000 3500 4000 Frequency (Hz) 3:A>
09.,B.("C(+&(DE*,*8(F"2G A time-frequency plot can be obtained using the MATLAB s spectrogram function (Detail comes later Chapter) XX#%)-&*'./'UAQ89]?:8Y:8ST8H:::B EF"7JJ@7MN78./O4$"/)457,"$4(/'7'),&/3 EF"7F(3F",7F4,$&/(0'784O"7(/7)($" Listening to Tones To play the tuning fork sound on the PC speakers using Matlab we type XX#%.30AQ8H:::B where the second argument sets the sampling frequency for playback 3:A@
Time Signals: More Than Formulas D,#&(7,8*-%.H(I"2&(D-*(F"2#E%-. The signal modeling of this chapter has focused on single sinusoids In practice real signals are far more complex, even a multiple sinusoids model is only an approximation Modeling still has great value in system design 3:;6