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Chapter 2 READING ASSIGNMENTS This Lecture: • Chapter 2, pp. 9-17 • Appendix A: Complex Numbers • Appendix B: MATLAB or Labview • Chapter 1: Introduction LECTURE OBJECTIVES Write general formula for a sinusoidal” waveform, or signal From the formula, plot the sinusoid versus time What’s a signal? - It’s a function of time, x(t) - in the mathematical sense Sinusoidal is a general class of signals have simple mathematical representations They are the most basic signals in the theory of signals and systems it is important to become familiar with their properties. The most general mathematical formula for a cosine signal is Acos(ω t+ϕ) Example 1. Note that x (t) oscillates between A and —A, and that it repeats the same pattern of oscillations every 1 /440 = 0.00227 sec (approximately). This time interval is called the period of the sinusoid. How to verify your answer with LabVIEW? Recording of tuning fork TUNING FORK EXAMPLE 400 Hz Review of Sine and Cosine Functions Basic properties of the sine and cosine functions Sinusoidal Signals A is called the amplitude . The amplitude is a scaling factor that determines how large the cosine signal will be. Since the function cos oscillates between +1 and — 1, is called the phase shift The units of phase shift must be radians Refer to page 11 , see how to obtain the following results 0 is called the radian frequency. Notes • Angles are therefore specified in radians. • If the angle is in the first quadrant (0 < < /2 rad), then the sine of is the length y of the side of the triangle opposite the angle divided by the length r of the hypotenuse of the right triangle. • as increases from 0 to /2, cos decreases from 1 to 0 and sin increases from to 0 to 1 Sine and cosine functions plotted versus angle 9. Both functions have a period of 2. Some basic trigonometric identities The main parameters of Sinusoidal Signals • A is called the amplitude . The amplitude is a scaling factor that determines how large the cosine signal will be. • is called the phase shift. The units of phase shift must be radians, since the argument of the cosine must be in radians. PLOTTING COSINE SIGNAL from the FORMULA Relation of Frequency to Period the sinusoid determines its period, and the relationship can be found by examining the following equations: Since the cosine function has a period of 2, the equality above holds for all values of t if Cosine signals x(t) = 5cos(2f0t) for several values of f0: Phase Shift and Time Shift • The phase shift parameter (together with the frequency) determines the time locations of the maxima and minima of a cosine wave. • The phase shift determine how much the maximum of cos or sin is shifted away fro t=0; Example: sinusoidal signal has a positive peak at t=0; when 0 Example x(t)=s(t) This simple function has • a positive slope of 2 for 0t ½ • a negative slope of -3/2 for ½ <t 3 Example x(t)=s(t-2) Example x(t)=s(t+1) x(t)= ? Exercise: Derive the equations for the shifted signal x2 (t) = s(t + 1). x2(t)=s(t+1) Slope=-1/2 Slope=1 -1 1 𝑡 + 1 , −1 ≤ 𝑡 ≤ 0 1 𝑥2 𝑡 = 1 − 𝑡 , 0 ≤ 𝑡 ≤ 2 2 0, 𝑜𝑡ℎ𝑒𝑟𝑤ℎ𝑒𝑟𝑒 2 Under standing Time-shifting A Signal Whenever a signal can be expressed in the form x1 (t) = s(t – t1 ) we say that x1 (t) is a time-shifted version of s(t). If t1>0, then the shift is to the right, and we say that the signal s(t) has been delayed in time. If t1<0 , then the shift is to the left, and we say that the signal s(t) was advanced in time If t1=0 ? TIME-SHIFT In mathematical formula we can replace t with t-tm Then the t=0 point moves to t=tm Peak value of cos(ω(t-tm)) is now at t=tm Determination the time shift for a cosine signal The positive peak occurs is t= — 0.005 sec. Converting delay to phase shift Since this equation must hold for all t, we must have -0t0=, which leads to Notice that the phase shift is negative when the time shift is positive (a delay). In terms of the period (T0 = l/f0 ) we get the more intuitive formula the positive peak nearest to t =0 must always lie within |t1| T0/2 the phase shift can always be chosen to satisfy — the phase shift is also ambiguous because adding a multiple of 2 to the argument of a cosine function does not change the value of the cosine. EXERCISE 2.3: In the Fig. 2-6, it is possible to measure both a positive and a negative value of t1 and then calculate the corresponding phase shifts. Which phase shift is within the range — ? Verify that the two phase shifts differ by 2 . PLOTTING COSINE SIGNAL from the FORMULA Given the following formula SINUSOID from a PLOT Example : SINUSOID from a PLOT Sampling and Plotting Sinusoids where Ts is called the sample spacing or sampling period, and n is an integer. n = -7:5; Ts = 0.005; tn = n*Ts; xn = 20*cos (80*pi*tn - 0.4*pi); plot (tn,xn) Complex Exponentials and Phasors Cartesian and polar representations of complex numbers in the complex plane. COMPLEX NUMBERS Inverse Euler Formulas Application EXERCISE Show that the following representation can be derived for the real sine signal: