ENGR 4323/5323 Digital and Analog Communication Ch2 Signals and Signal Space Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr Outline • • • • • • • Size of a Signal Classification of Signals Useful Signals and Signal Operations Signals Versus Vectors Correlation of Signals Orthogonal Signal Sets Trigonometric and Exponential Fourier Series Signal Energy and Power Energy Signal Power Signal ∞ ∞ Energy Power 𝐸𝑔 = 𝑔2 𝑡 𝑑𝑡 −∞ 1 𝑃𝑔 = lim 𝑇→∞ 𝑇 𝑔(𝑡) 2 𝑑𝑡 𝐸𝑔 = −∞ 𝑇/2 𝑔2 𝑡 𝑑𝑡 −𝑇/2 1 𝑃𝑔 = lim 𝑇→∞ 𝑇 𝑇/2 |𝑔 𝑡 |2 𝑑𝑡 −𝑇/2 Signal Classification 1. Continuous-time and discrete-time signals 2. Analog and digital signals 3. Periodic and aperiodic signals 4. Energy and power signals 5. Deterministic and probabilistic signals 6. Causal and non-causal 7. Even and Odd signals Analog continuous Digital continuous Analog Discrete Digital Discrete Periodic Deterministic Aperiodic Probabilistic Useful Signal Operation Time Shifting Time Scaling Time Inversion 2 1 -1 1 Useful Signals Unit impulse Signal Unit step function u(t) 𝑑𝑢(𝑡) = 𝛿(𝑡) 𝑑𝑡 Signals Versus Vectors By sampling, a continuous signal g(t) can be represented as vector g. g = [ g(t1) g(t2) … g(tn)] Vector Approximation To approximate vector g using another vector x then we need to choose c that will minimize the error e. g = cx + e Dot product: <g, x> = ||g||.||x|| cos θ Signals Versus Vectors Value of c that minimizes the error Signal Approximation g(t) = cx(t) + e(t) x(t) Correlation of Signals Two vectors are similar if the angle between them is small. Correlation coefficient Note: Similarity between vectors or signals does not depend on the length of the vectors or the strength of the signals. Example Which of the signals g1(t), g2(t), …, g6(t) are similar to x(t)? 1=1 2=1 3=-1 4=0.961 5=0.628 6=0 Cross-Correlation Function 4 g(t-) Autocorrelation Function g(t) g(t-) Orthogonal Signal Sets Orthogonal Vector Space g = c1x1 + c2x2 + c3x3 Orthogonal Signal Space g(t) = c1x1(t)+ c2x2(t) + … + cNxN(t) Parseval’s Theorem Trigonometric Fourier Series n 1 n 1 xt a0 an cos2nf 0t bn sin 2nf 0t 1 a0 xt dt T0 T0 2 an xt cos2nf 0t dt T0 T0 2 bn xt sin 2nf 0t dt T0 T0 Example f t a0 an cos2nt bn sin 2nt f(t) 1 n 1 e-t/2 0 • Fundamental period T0 = • Fundamental frequency f0 = 1/T0 = 1/ Hz w0 = 2/T0 = 2 rad/s a0 an 1 2 0 0 2 2 e dt e 1 0.504 e t 2 t 2 2 cos2nt dt 0.504 2 1 16n 8n bn e sin 2nt dt 0.504 2 0 1 16n an and bn decrease in amplitude as n . 2 t 2 2 8n f t 0.504 1 cos2nt sin 2nt 2 2 1 16n n 1 1 16n To what value does the FS converge at the point of discontinuity? Compact Trigonometric Fourier Series We can use the trigonometric identity a cos(x) + b sin(x) = c cos(x + ) to find the compact trigonometric Fourier series xt C0 dc component Cn cos2nf 0t n n 1 nth harmonic C0, Cn, and θn are related to the trigonometric coefficients an and bn as: C0 a0 Cn an bn 2 2 bn n tan an 1 Role of Amplitude in Shaping Waveform xt C0 Cn cos2nf 0t n n 1 Role of the Phase in Shaping a Periodic Signal xt C0 Cn cos2nf 0t n n 1 Compact Trigonometric f t C0 Cn cos2nt n f(t) n 1 1 a0 0.504 e-t/2 0 2 an 0.504 2 1 16n 8n bn 0.504 2 1 16n C0 ao 0.504 • Fundamental period T0 = • Fundamental frequency f0 = 1/T0 = 1/ Hz w0 = 2/T0 = 2 rad/s 2 Cn a b 0.504 2 1 16n 2 n 2 n bn tan 1 4n n tan an 1 f t 0.504 0.504 n 1 2 1 16n 2 cos 2nt tan 1 4n Line Spectra of x(t) • The amplitude spectrum of x(t) is defined as the plot of the magnitudes |Cn| versus w • The phase spectrum of x(t) is defined as the plot of the angles Cn phase(Cn ) versus w • This results in line spectra • Bandwidth the difference between the highest and lowest frequencies of the spectral components of a signal. Line Spectra f(t) e-t/2 0 2 Cn 0.504 2 1 16n C0 0.504 1 f t 0.504 0.504 n 1 n tan 1 4n 2 cos 2nt tan 1 4n 1 16n 2 f(t)=0.504 + 0.244 cos(2t-75.96o) + 0.125 cos(4t-82.87o) + o) + 0.063 cos(8t-86.24o) + … 0.084 cos(6t-85.24 C n n 0.504 0.244 w 0.125 0.084 0 2 4 6 0.063 8 10 w -/2 Exponential Fourier Series xt D e n j 2f 0 nt n j 2f 0 nt To find Dn multiply both side by e over a full period, m =1,2,…,n,… 1 Dn To and then integrate j 2f 0 nt x t e dt , n 0, 1, 2,.... To Dn is a complex quantity in general Dn=|Dn|ej D-n = Dn* |Dn|=|D-n| Even Dn = - D-n Odd D0 is called the constant or dc component of x(t) Line Spectra in the Exponential Form • The line spectra for the exponential form has negative frequencies because of the mathematical nature of the complex exponent. x(t ) ... | D 2 | e j 2 e j 2w0t | D1 | e j1 e jw0t D0 | D1 | e j1 e jw0t | D2 | e j 2 e j 2w0t ... x(t ) C0 C1 cos(w0t 1 ) C2 cos(2w0t 2 ) ... D0= C0 Cn = 2 |Dn| Dn = n0 D-n = - Cn Cn Example Find the exponential Fourier Series for the squaref(t) pulse periodic signal. 1 Dn 2 1 /2 jnt e dt / 2 sin n / 2 0.5sinc( n / 2) n 1 D0 2 n even 0 Dn 1 / n n odd 0 n for all n 3,7,11,15, n 3,7,11,15, 2 /2 /2 2 • Fundamental period T0 = 2 • Fundamental frequency f0 = 1/T0 = 1/2 Hz w0 = 2/T0 = 1 rad/s Exponential Line Spectra |Dn| 1 1 Dn 1 1 Example The compact trigonometric Fourier Series coefficients for the square-pulse periodic signal. f(t) 1 C0 2 0 n even Cn 2 n odd n 0 for all n 3,7,11,15, n n 3,7,11,15, 1 2 /2 /2 2 • Fundamental frequency f0 = 1/T0 = 1/2 Hz w0 = 2/T0 = 1 rad/s 1 2 ( n 1) / 2 f (t ) cos nt (1) 1 2 n 1 n 2 Relationships between the Coefficients of the Different Forms an Dn D n 2 ReDn 2 2 bk j Dn D n 2 ImDn Cn an bn an Cn cos n bn Cn sin n 1 bn n tan a0 D0 c0 an Dn 0.5an jbn D n D n 0.5an jbn Dn 0.5Cn n 0.5Cn e D0 a0 C0 j n Cn 2 Dn n Dn C0 a0 D0 Example Find the exponential Fourier Series and sketch the corresponding spectra for the impulse train shown below. From this result sketch the trigonometric spectrum and write the trigonometric Fourier Series. Solution T (t ) Dn 1 / T0 1 T0 (t ) T0 0 jnw0t e n Cn 2 | Dn | 2 / T0 C0 | D0 | 1 / T0 1 T0 (t ) T0 1 2 cos( nw0t ) n 1 -2T0 -T0 T0 2T0 Dirichlet Conditions for FS Convergence 1- The function g(t) is absolutely integrable over one period. 𝑇0 𝑔 𝑡 𝑑𝑡 < ∞ 2- The function g(t) can have only a finite number of maxima, minima, and discontinuities in one period. Parseval’s Theorem • Let x(t) be a periodic signal with period T • The average power P of the signal is defined as 1 P T T /2 T / 2 2 x(t ) dt • Expressing the signal as xt C0 Cn cos( nw0t n ) n 1 it is also P C0 0.5Cn 2 n 1 2 P D 2 Dn 2 0 n 1 2 Numerical Calculation of Fourier Series 1 Dn To j 2f 0 nt g t e dt , n 0, 1, 2,.... To 1 𝐷𝑛 = lim 𝑇𝑠 →0 𝑇0 𝑁0 −1 1 𝐷𝑛 = lim 𝑇𝑠 →0 𝑁0 1 𝐷𝑛 = 𝑁0 𝑔 𝑘𝑇𝑠 𝑒 −𝑗𝑛𝜔0 𝑘𝑇𝑠 𝑇𝑠 𝑘=0 𝑁0 −1 𝑔 𝑘𝑇𝑠 𝑒 −𝑗𝑛Ω0𝑘 𝑘=0 𝑁0 −1 𝑔 𝑘𝑇𝑠 𝑒 −𝑗𝑛Ω0 𝑘 𝑘=0 Ω0 = 𝜔0 𝑇𝑠 𝑇0 𝑁0 = 𝑇𝑠 Matlab Exercise Page 80 g(t) 1 T0 = pi; e N0 = 256; 0 Ts = T0/N0; t = 0: Ts : Ts*(N0-1); T0 = g = exp(-t/2); f0 = 1/T0 = 1/ Hz g(1) = 0.604; w0 = 2/T0 = 2 rad/s Dn = fft(g)/N0; [Dnangle, Dnmag] = cart2pol( real(Dn), imag(Dn)); k =0:length(Dn)-1; k = 2*k; subplot(211), stem(k, Dnmag) subplot(212), stem(k,Dnangle) -t/2