SINUSOIDAL SIGNALS SINUSOIDAL SIGNALS x(t) = cos(2¼ft + ') = cos(!t + ') (continuous time) x[n] = cos(2¼fn + ') = cos(!n + ') (discrete time) f : frequency (s¡1 (Hz)) ! : angular frequency (radians/s) ' : phase (radians) 2 Sinusoidal signals A cos(ω t) A sin(ω t) A 0 −A 0 T/2 T 3T/2 2T xc(t) = A cos(2¼ft) xs(t) = A sin(2¼ft) = A cos(2¼ft ¡ ¼=2) - amplitude A -amplitude A - period T = 1=f -period T = 1=f - phase: 0 -phase: ¡¼=2 3 WHY SINUSOIDAL SIGNALS? ² Physical reasons: - harmonic oscillators generate sinusoids, e.g., vibrating structures - waves consist of sinusoidals, e.g., acoustic waves or electromagnetic waves used in wireless transmission ² Psychophysical reason: - speech consists of superposition of sinusoids - human ear detects frequencies - human eye senses light of various frequencies ² Mathematical (and physical) reason: - Linear systems, both physical systems and man-made ¯lters, a®ect a signal frequency by frequency (hence low- pass, high-pass etc ¯lters) 4 EXAMPLE: TRANSMISSION OF A LOW-FREQUENCY SIGNAL USING HIGH-FREQUENCY ELECTROMAGNETIC (RADIO) SIGNAL - A POSSIBLE (CONVENTIONAL) METHOD: AMPLITUDE MODULATION (AM) Example: low-frequency signal v(t) = 5 + 2 cos(2¼f¢t); f¢ = 20 Hz High-frequency carrier wave vc(t) = cos(2¼fct); fc = 200 Hz Amplitude modulation (AM) of carrier (electromagnetic) wave: x(t) = v(t) cos(2¼fct) 5 8 6 4 2 v 0 −2 −4 −6 −8 8 6 4 2 x 0 −2 −4 −6 −8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t Top: v(t) (dashed) and vc(t) = cos(2¼fct). Bottom: transmitted signal x(t) = v(t) cos(2¼fct). 6 Frequency contents of transmitted signal x(t) = v(t) cos(2¼fct) = (5 + 2 cos(2¼f¢t)) cos(2¼fct) = 5 cos(2¼fct) + 2 cos(2¼f¢t) cos(2¼fct) Trigonometric identity: 1 1 cos ® cos ¯ = cos(® ¡ ¯) + cos(® + ¯) 2 2 ) 2 cos(2¼f¢t) cos(2¼fct) = cos (2¼(fc ¡ f¢)t)+cos (2¼(fc + f¢)t) 7 ) x(t) = 5 cos(2¼fct) + cos (2¼(fc ¡ f¢)t) + cos (2¼(fc + f¢)t) 5 Spectrum Spectrum of v(t) of vc(t) 2 1 - 0 20 200 Frequency Spectrum of x(t) 5 1 1 - 0 180 200 220 Frequency 8 We see that a low-frequency signal in frequency range 0 · fs · fmax (baseband signal) can be transmitted as a signal in the frequency range fc ¡ fmax · f · fc ¡ fmax ("RF" (radio frequency) signal). Another analog modulation technique is frequency modulation (FM) 9 Representation of sinusoidal signal A. Amplitude and phase B. Sine and cosine components C. Complex exponentials 10 A. Amplitude and phase x(t) = A cos(2¼ft + ') or x(t) = A cos(!t + ') where ! = 2¼f is the angular frequency (radians/second) The phase ' describes translation in time compared to cos(!t): x(t) = A cos(!t + ') = A cos(!(t + '=!)) 11 B. Sine and cosine components x(t) = a cos(!t) + b sin(!t) Follows from trigonometric identity: cos(® + ¯) = cos(®) cos(¯) ¡ sin(®) sin(¯) ) x(t) = A cos(!t + ') = A cos(') cos(!t) ¡ A sin(') sin(!t) Hence: a = A cos('); b = ¡A sin(') and, conversely, p A = a2 + b2;' = ¡ arctan(b=a) 12 EXAMPLE x(t) 1 0 −1 −2 0 2 4 6 8 10 12 Signal x(t) = cos(!t + ') Period: T = 10 (s) Frequency: f = 1=T = 0:1 (s¡1 =Hz (periods/second)) Angular frequency: ! = 2¼f = 0:628 (radians/s) 13 A. Representation using amplitude and phase Amplitude: A = 1 From ¯gure we see that x(t) = cos(!(t ¡ 2)). But x(t) = cos(!t + ') = cos(!(t + '=!) ) Phase: ' = ¡2! = ¡0:4¼ radians or ¡0:4 ¢ 180=¼ = ¡72± 14 B. Sine and cosine components x(t) = a cos(!t) + b sin(!t) where a = cos(') = cos(¡0:4¼) = 0:3090 b = ¡ sin(') = ¡ sin(¡0:4¼) = 0:9511 1 x(t) 0 −1 −2 0 2 4 6 8 10 12 1 Cosine component Sine component 0 −1 −2 0 2 4 6 8 10 12 x(t) = 0:3090 cos(!t) + 0:9511 sin(!t) 15 C. Representation using complex exponentials Background. A common operation is to combine sinusoidal components with di®erent phases: x(t) = A1 cos(!t + '1) + A2 cos(!t + '2) One way: ¯rst write A1 cos(!t + '1) = a1 cos(!t) + b1 sin(!t) A2 cos(!t + '2) = a2 cos(!t) + b2 sin(!t) Then x(t) = (a1 + a2) cos(!t) + (b1 + b2) sin(!t) 16 Streamlining the above procedure We have: cos(!t + ') = cos(') cos(!t) ¡ sin(') sin(!t) and sin(!t + ') = sin(') cos(!t) + cos(') sin(!t) Instead of working with the two signals cos(!t) and sin(!t) it is convenient to working with the single complex-valued signal g(!t) = cos(!t) + j sin(!t) p where j = ¡1 (imaginary unit number) 17 Im 6 g(!t) = cos(!t) + j sin(!t) j sin(!t) ©©* ©© ©© ©© 1 ©© ©© ©© ©© ©© ©© !t ©© - cos(!t) Re Then: g(!t + ') = cos(!t + ') + j sin(!t + ') = cos(') cos(!t) ¡ sin(') sin(!t) ¡ ¢ +j sin(') cos(!t) + cos(') sin(!t) ¡ ¢¡ ¢ = cos(') + j sin(') cos(!t) + j sin(!t) = g(')g(!t) 18 In terms of the complex-valued signal, phase shift with angle ' is equivalent to multiplication by the complex number g('): g(!t + ') = g(')g(!t)(A) ) cos(!t + ') = Re (g(')g(!t)) sin(!t + ') = Im (g(')g(!t)) We will see that using the multiplication (A) to describe phase shifts results in signi¯cant simpli¯cation of formulae and calculations 19 Properties of the complex function g(θ) = cos(θ) + j sin(θ): - g(0) = 1 - g(θ1 + θ2) = g(θ1)g(θ2) dg(θ) - dθ = ¡ sin(θ) + j cos(θ) = jg(θ) The only function which satis¯es these relations is g(θ) = ejθ e = 2:71828182845904523536 ::: (Euler's number) 20 Hence we have: Euler's relation ejθ = cos(θ) + j sin(θ) e = 2:71828182845904523536 ::: (Euler's number) As e¡jθ = cos(θ) ¡ j sin(θ) we have 1 ¡ ¢ cos(θ) = ejθ + e¡jθ 2 1 ¡ ¢ sin(θ) = ejθ ¡ e¡jθ 2j 21 It follows that x(t) = a cos(!t) + b sin(!t) µ ¶ µ ¶ a b a b = + ej!t + ¡ e¡j!t 2 2j 2 2j 1 1 = (a ¡ jb) ej!t + (a + jb) e¡j!t 2 2 or x(t) = c ej!t + c¤e¡j!t 1 ¤ 1 where c = 2 (a ¡ jb) and c = 2 (a + jb) (complex conjugate) Thus the cosine and sine components are parameterized as the real and imaginary components of the complex number c 22 PREVIOUS EXAMPLE: The signal x(t) = cos(!t ¡ 0:4¼) = a cos(!t) + b sin(!t) = 0:3090 cos(!t) + 0:9511 sin(!t) can be represented using complex exponentials as x(t) = c ej!t + c¤ e¡j!t where 1 c = (a ¡ jb) = 0:1545 ¡ j0:4755 2 1 c¤ = (a + jb) = 0:1545 + j0:4755 2 Hence: x(t) = (0:1545 ¡ j0:4755) ej!t + (0:1545 + j0:4755) e¡j!t 23 SUMMARY Let us summarize the results so far. We have the following alternative representations of a sinusoidal signal: ² Amplitude and phase: x(t) = A cos(!t + ') ² Sine and cosine components: x(t) = a cos(!t) + b sin(!t) where a = A cos('); b = ¡A sin(') or p A = a2 + b2;' = ¡ arctan(b=a) 24 ² Complex exponentials: x(t) = c ej!t + c¤ e¡j!t where 1 1 c = (a ¡ jb) ; c¤ = (a + jb) 2 2 or a = 2Re(c); b = ¡2Im(c) 25 EXAMPLE - Signal transmission with reflected component Transmitted signal: x(t) = cos(!t) Received signal: y(t) = x(t) + rx(t ¡ ¿) PROBLEM: Determine amplitude and phase of y(t) We have y(t) = cos(!t) + r cos(!(t ¡ ¿)) The problem can be solved in a straightforward way by introducing complex exponential: 1 ¡ ¢ x(t) = ej!t + e¡j!t 2 26 Then: y(t) = cos(!t) + r cos(!(t ¡ ¿)) 1 ¡ ¢ 1 ³ ´ = ej!t + e¡j!t + r ej!(t¡¿) + e¡j!(t¡¿) 2 2 1 ¡ ¢ 1 ¡ ¢ = 1 + re¡j!¿ ej!t + 1 + rej!¿ e¡j!t 2 2 Here, 1 ¡ ¢ 1³ ´ 1 + re¡j!¿ = 1 + r cos(!¿) ¡ jr sin(!¿) 2 2 and 1 ¡ ¢ 1³ ´ 1 + rej!¿ = 1 + r cos(!¿) + jr sin(!¿) 2 2 27 Then, de¯ning a = 1 + r cos(!¿); b = r sin(!¿) we have (cf. above) 1 1 y(t) = (a ¡ jb) ej!t + (a + jb) e¡j!t 2 2 = a cos(!t) + b sin(!t) = Ay cos(!t + 'y) where p 2 2 Ay = a + b ;'y = ¡ arctan(b=a) 28 SOME SIMPLIFICATIONS As 1 1 a cos(!t) + b sin(!t) = (a ¡ jb) ej!t + (a + jb) e¡j!t 2 2 it follows that it is su±cient to consider the complex-valued signal 1 (a ¡ jb) ej!t 2 Here, the complex number 1 c = (a ¡ jb) 2 determines the signal's amplitude and phase. 29 To see how this works, consider a linear dynamical system with input x(t) and output y(t) x(t) - G - y(t) For the sinusoidal input x(t) = cos(!t) the output takes the form y(t) = A cos(!t + ') = a cos(!t) + b sin(!t) and we want to determine the amplitude A and phase '. 30 As the phase shift introduces a combination of cosines and sines, the problem can be simpli¯ed by embedding the input signal into a larger class of signals involving both a cosine and a sine component. It turns out that a convenient way to do this is to consider the complex input signal xc(t) = cos(!t) + j sin(!t) = ej!t The output then also has real and imaginary components: yc(t) = A cos(!t + ') + jA sin(!t + ') = Aej(!t+') = Aej'ej!t 31 or j!t yc(t) = c e = c xc(t) where c = Aej'.