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(2f0t) 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 0t  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: