# Equation for a line. Synthetic Impulse Response Time (sec) x(t) m

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1 Fundamenals of Signals Overview Definiion Examples Energy and power Signal ransformaions Periodic signals Symmery Exponenial & sinusoidal signals Basis funcions Equaion for a line x() m x() =m( ) You will ofen need o quickly wrie an expression for a line given he slope and x-inercep Will use ofen when discussing convoluion and Fourier ransforms You should know how o apply his J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 2 Examples of Signals Definiion: an absracion of any measurable quaniy ha is a funcion of one or more independen variables such as ime or space. Examples: A volage in a circui A curren in a circui The Dow Jones Indusrial average Elecrocardiograms A sin(ω + φ) Speech/music Force exered on a shock absorber Concenraion of Chlorine in a waer supply Synheic Impulse Response Time (sec) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 3 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 4

2 Microelecrode Recording Elecrocardiogram Time (sec) Time (sec) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 5 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 6 Arerial Blood Pressure Speech ABP (mmhg) Linus: Philosophy of We Suckers Time (sec) Time (sec) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 7 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 8

3 Chaos Time (samples) Discree-ime & Coninuous-ime We will work wih boh ypes of signals Coninuous-ime signals Will always be reaed as a funcion of Parenheses will be used o denoe coninuous-ime funcions Example: x() is a coninuous independen variable (real-valued) Discree-ime signals Will always be reaed as a funcion of n Square brackes will be used o denoe discree-ime funcions Example: x[n] n is an independen ineger J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 9 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 Signal Energy & Power For mos of his class we will use a broad definiion of power and energy ha applies o any signal x() or x[n] Insananeous signal power P () = x() 2 P [n] = x[n] 2 Signal energy E(, )= x() 2 d E(n,n )= Average signal power P (, )= P (n,n )= n n + x() 2 d n n n=n x[n] 2 n=n x[n] 2 Signal Energy & Power Commens Usually, he limis are aken over an infinie ime inerval E = x() 2 d E = x[n] 2 T P = lim x() 2 d T 2T T n= P = lim N 2N + N n= N x[n] 2 We will encouner many ypes of signals Some have infinie average power, energy, or boh A signal is called an energy signal if E < A signal is called a power signal if <P < A signal can be an energy signal, a power signal, or neiher ype A signal can no be boh an energy signal and a power signal J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 2

4 Example : Energy & Power Deermine wheher he energy and average power of each of he following signals is finie. 8 < 5 x() = oherwise x[n] =j x[n] =A cos(ωn + φ) e a > x() = oherwise x[n] =e jωn Example : Workspace () J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 3 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 4 Example : Workspace (2) Signal Energy & Power Tips There are a few rules ha can help you deermine wheher a signal has finie energy and average power Signals wih finie energy have zero average power: E < P = Signals of finie duraion and ampliude have finie energy: x() =for >c E < Signals wih finie average power have infinie energy: P > E = J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 5 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 6

5 Signal Transformaions Example 2: Signal Transformaions Time shif: x( ) and x[n n ] If > or n >, signal is shifed o he righ x() If < or n <, signal is shifed o he lef Time reversal: x( ) and x[ n] - Time scaling: x(α) and x[αn] If α>, signal appears compressed If >α>, signal appears sreched Use he signal shown above o draw he following: x( ), x( ), x( +2), x( 2 ), x(2), x(2 2). J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 7 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 8 Example 2: Axes for x( +2) & x( 2 ) Example 2: Axes for x(2) & x(2 2) x() x() J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 9 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 2

6 Even & Odd Symmery x e () = 2 (x()+x( )) x o () = 2 (x() x( )) x() = x e ()+x o () The symmery of a signal under ime reversal will be useful laer when we discuss ransforms A signal is even if and only if x() =x( ) A signal is odd if and only if x() = x( ) cos(kω ) is an even signal sin(kω ) is an odd signal Any signal can be wrien as he sum of an odd signal and an even signal Example 3: Even Symmery x() Draw he even componen of he signal shown above. J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 2 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver Example 4: Odd Symmery Example 5: Even & Odd Symmery x() Draw he odd componen of he signal shown above. Show ha he sum of he even and odd componens of he signal is equal o he original signal graphically. J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 24

7 Periodic Signals A signal is periodic if here is a posiive value of T or N such ha x() =x( + T ) x[n] =x[n + N] The fundamenal period,t, for coninuous-ime signals is he smalles posiive value of T such ha x() =x( + T ) The fundamenal period,n, for discree-ime signals is he smalles posiive ineger of N such ha x[n] =x[n + N] Signals ha are no periodic are said o be aperiodic Exponenial and Sinusoidal Signals Exponenial signals x() =Ae a x[n] =Ae an where A and a are complex numbers. Exponenial and sinusoidal signals arise naurally in he analysis of linear sysems Example: simple harmonic moion ha you learned in physics There are several disinc ypes of exponenial signals A and a real A and a imaginary A and a complex (mos general case) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver Example 6: Ae an, A =and a = ± Example 6: MATLAB Code n = -:3; % Time index subplo(2,,); y = exp(n/5); % Growing exponenial h = sem(n,y); se(h(), Marker,. ); se(gca, Box, Off ); subplo(2,,2); y = exp(-n/5); % Decaying exponenial h = sem(n,y); se(h(), Marker,. ); se(gca, Box, Off ); xlabel( Time (n) ); Time (n) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 28

8 Sinusoidal Exponenial Signal Commens Example 7: Ae a, A =and a = j x() =Ae a = A(e a ) = Aα x[n] =Ae an = A(e a ) n = Aα n Complex:Blue Real:Red Imaginary:Green When a is imaginary, hen Euler s equaion applies: e jω = cos(ω)+j sin(ω) e jωn = cos(ωn)+j sin(ωn) Since e jω =, his looks like a coil in a plo of he complex plane versus ime e jω is Periodic wih fundamenal period T = 2π ω Real par is sinusoidal: ReAe jω } = A cos(ω) Imaginary par is sinusoidal: ImAe jω } = A sin(ω) These signals have infinie energy, bu finie (consan) average power, P Real Par.5.5 Time (s) 2 3 Imaginary Par J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 3 Example 7: MATLAB Code fs = 5; % Sample rae (Hz) = -:/fs:3; % Time index (s) a = j; A = ; y = A*exp(a*); h = plo3(,imag(y),real(y), b ); hold on; h = plo3(,ones(size()),real(y), r ); h = plo3(,imag(y),-ones(size()), g ); hold off; grid on; xlabel( Time (s) ); ylabel( Imaginary Par ); zlabel( Real Par ); ile( Complex:Blue Real:Red Imaginary:Green ); view(27.5,22); Sinusoidal Exponenial Harmonics In order for e jω o be periodic wih period T, we require ha e jω =e jω(+t ) =e jω e jωt for all This implies e jωt =and herefore ωt =2πk where k =, ±, ±2,... There is more han one frequency ω ha saisfies he consrain x() =x( + T ) where T = 2πk ω The fundamenal frequency is given by k =: ω = 2π T The oher frequencies ha saisfy his consrain are hen ineger muliples of ω J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 3 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 32

9 Sinusoidal Exponenial Harmonics Coninued A harmonically relaed se of complex exponenials is a se of exponenials wih fundamenal frequencies ha are all muliples of a single posiive frequency ω φ k () =e jkω where k =, ±, ±2,... For k =, φ k () is a consan For all oher values φ k () is periodic wih fundamenal frequency k ω This is consisen wih how he erm harmonic is used in music Sinusoidal harmonics will play a very imporan role when we discuss Fourier series and periodic signals φ φ φ 2 φ 3 φ 4 Example 8: Coninuous-Time Harmonics Time (s) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver φ φ φ 2 φ 3 φ 4 Example 9: Discree-Time Harmonics Time (n) Example 9: MATLAB Code n = -:4; % Time index omega = 2*pi/2; % Frequency (radians/sample) N = 4; for cn = :N, subplo(n+,,cn+); phi = exp(j*cn*omega*n); h = sem(n,real(phi)); se([h()], Marker,. ); ylim([-..]); ylabel(sprinf( \\phi_%d,cn)); se(gca, YGrid, On ) if cn~=n, se(gca, XTickLabel,[]); end; end; xlabel( Time (n) ); figure; = -:.:4; % Time index omega =.5*2*pi; % Frequency (radians/sec) -.5 Hz N = 4; for cn = :N, subplo(n+,,cn+); phi = exp(j*cn*omega*); h = plo(,real(phi)); ylim([-..]); ylabel(sprinf( \\phi_%d,cn)); grid on; if cn~=n, se(gca, XTickLabel,[]); end; end; xlabel( Time (s) ); J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 36

10 Damped Complex Sinusoidal Exponenials Example : Ae an, A =and a = ±.+j.5 x() =Ae a x[n] =Ae an 5 When a is complex, hese become damped sinusoidal exponenials Le a = α + jω. Then x() =Ae a =(Ae α ) e jω x[n] =Ae an =(Ae αn ) e jωn Thus, hese are equivalen o muliplying an complex sinusoid by a real exponenial Real Par Real Par Time (n) J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver Example : MATLAB Code Example : Ae a, A =and a = ±.5 + j2 n = -:4; % Time index C = ; subplo(2,,); a =. + j*.5; y = real(c*exp(a*n)); % Growing exponenial = min(n):.:max(n); h = plo(, real(c*exp(real(a)*)),,-real(c*exp(real(a)*))); se(h, Color,[.5.5.5]); hold on; h = sem(n,y); se(h(), Marker,. ); hold off; box off; grid on; ylim([-5 5]); ylabel( Real Par ); subplo(2,,2); a = -. + j*.5; y = real(c*exp(a*n)); % Growing exponenial = min(n):.:max(n); h = plo(, real(c*exp(real(a)*)),,-real(c*exp(real(a)*))); se(h, Color,[.5.5.5]); hold on; h = sem(n,y); se(h(), Marker,. ); hold off; box off; grid on; ylim([ 2]); xlabel( Time (n) ); ylabel( Real Par ); Real Par Real Par Complex:Blue Real:Red Imaginary:Green Time (s) 2 Imaginary Par 2 Imaginary Par J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 4

11 Example : MATLAB Code fs = 5; % Sample rae (Hz) = -:/fs:3; % Time index (s) subplo(2,,); a = j*2; y = C*exp(a*); h = plo3(,imag(y),real(y), b ); hold on; h = plo3(,2*ones(size()),real(y), r ); h = plo3(,imag(y),*ones(size()), g ); hold off; grid on; ylabel( Imaginary Par ); zlabel( Real Par ); ile( Complex:Blue Real:Red Imaginary:Green ); axis([min() max() 2 2]); view(27.5,22); subplo(2,,2); a = j*2; y = C*exp(a*); h = plo3(,imag(y),real(y), b ); hold on; h = plo3(,2*ones(size()),real(y), r ); h = plo3(,imag(y),*ones(size()), g ); hold off; grid on; xlabel( Time (s) ); ylabel( Imaginary Par ); zlabel( Real Par ); axis([min() max() 2 2]); view(27.5,22); Discree-Time Uni Impulse The discree-ime uni impulse is defined as, n δ[n] =, n = Someimes called he uni sample Also called he Kroneker dela Noe ha δ[n] has even symmery so δ[n] =δ[ n] n J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 4 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver Discree-Time Uni Sep The discree-ime uni sep is defined as, n < u[n] =, n n Discree-Time Basis Funcions There is a close relaionship beween δ[n] and u[n] δ[n] = u[n] u[n ] n u[n] = δ[k] u[n] = k= δ[n k] k= The uni impulse can be used o sample a discree-ime signal x[n]: x[] = x[k]δ[k] x[n] = x[k]δ[n k] k= k= This abiliy o use he uni impulse o exrac a single value of x[n] hrough muliplicaion will play an imporan role laer in he erm J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 44

12 Coninuous-Time Uni Sep u() u() < > Someimes known as he Heaviside funcion Disconinuous a = u() is no defined No of consequence because i is undefined for an infiniesimal period of ime v s i s = = Uni Sep for Swiches Linear Circui Linear Circui v s u() i s u() Linear Circui Linear Circui u() useful for represening he opening or closing of swiches We will ofen solve for or be given iniial condiions a = We can hen represen independen sources as hough hey were immediaely applied a =. More laer. J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver u e () Coninuous-Time Uni Impulse -e e -e e u() δ() δ e () du e() d As e, u e () u() δ e () for =becomes very large δ e () for becomes zero δ() lim e δ e () δ e () Coninuous-Time Uni Impulse Coninued δ() δ() = e e δ()d =for any e> Also known as he Dirac dela funcion Is zero everywhere excep zero The impulse inegral serves as a measure of he impulse ampliude Drawn as an arrow wih uni heigh 5δ() would be drawn as an arrow wih heigh of 5 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 48

13 Coninuous-Time Uni Impulse Commens δ() = = The impulse should be viewed as an idealizaion Real sysems wih finie ineria do no respond insananeously The mos imporan propery of an impulse is is area Mos sysems will respond nearly he same o sharp pulses regardless of heir shape - if They have he same ampliude (area) Their duraion is much briefer han he sysem s response The idealized uni impulse is shor enough for any sysem Uni Impulse Properies δ()x() = δ()x() δ(a) = a δ() δ( ) = δ() δ() = du() d u() = δ(τ)dτ J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 5 Uni Impulse Sampling Propery + x()δ()d = + = x() = x() Similarly, + x()δ( )d = + + x()δ()d + = x( ) = x( ) δ()d x( )δ( )d δ( )d Uni Impulse Sampling Propery x() = + x(τ)δ(τ )dτ This inegral does no appear o be useful I will urn ou o be very useful I saes ha x() can be wrien as a linear combinaion of scaled and shifed uni impulses This will be a key concep when we discuss convoluion nex week J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 5 J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 52

14 Example 2: Coninuous-Time Uni-Ramp Example 3: Coninuous-Time Uni-Ramp Inegral r() r() Wha is he firs derivaive? r() Wha is he inegral of he uni ramp? J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver u() = Basis Funcion Relaionships du() d = δ() u(τ)dτ = r() δ(τ)dτ r() = dr() d = u() r(τ)dτ = 2 r()2 u(τ)dτ If we can wrie a signal x() in erms of u() and r(), iiseasyo find he derivaive Similarly, i is easy o inegrae Basis Funcions Translaed u(- ) δ(- ) r(- ) Can wrie simple expressions for he funcions ranslaed in ime Can scale he ampliude Any piecewise linear signal can be wrien in erms of basis funcions This makes i easy o calculae derivaives and inegrals Will no discuss how his erm Sufficien o know i can be done J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver J. McNames Porland Sae Universiy ECE 222 Signal Fundamenals Ver..5 56

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