Lesson_3 - EEL 3135 Discrete-Time Signals and Systems Dr Fred J Taylor Professor Lesson Title Sinusoids Lesson Number 03 Introduction Sinusoids are

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EEL 3135: Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: Sinusoids Lesson Number: 03 Introduction: Sinusoids are a fundamentally important class of signals. They are an essential element of modern electronic communication and control system operation. Collections of sinusoids can also be combined to synthesize complex signals. Sinusoidal waves can be produced by a number of mechanisms. Both electronic and mechanical constant frequency sinusoidal signals of frequency ϖ 0 can be produced by systems that can be modeled as a 2 nd order ordinary differential equation (ODE) of the form: ( 29 ( 29 t x dt t x d 0 2 2 ϖ = 1. In general, a sinusoidal signal can be mathematically modeled as: x(t)=Acos( φ (t)) 2. which logically begs the question, what is meant by frequency? Frequency is formally defined to be: ( 29 ( 29 dt t d t φ = 3. A familiar instance is the constant frequency case: φ (t)= ϖ 0 t+ θ 0 ; d 2 φ (t)/dt 2 = ϖ 0 4. Another time-dependent phase defined signal is the frequency modulation (FM) and phase modulation (PM). For example, and FM signal is given by φ (t)= ϖ 0 t+ x(t)dt and d 2 φ (t)/dt 2 = ϖ 0 + x(t). Interpreting Equation 2 as a constant frequency process yields the familiar equation: x(t)=Acos( ϖ 0 t+ θ 0 ) 5. where A is called the amplitude, ϖ 0 is the sinusoid’s frequency in radians/second (r/s), and θ 0 is the phase shift in radians or degrees. The period of the constant frequency signal, T 0 , is given by: T 0 =1/f 0 =2 π / ϖ 0 6. It should be realized that a sinusoidal time delay corresponds to a phase shift. This can be illustrated in the next example. 1
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EEL 3135: Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Example: Phase Shift Suppose x(t)=Acos( ϖ 0 t) is delayed by T 0 /4 seconds producing a signal y(t) = Acos( ϖ 0( t - T 0 /4))= Acos( ϖ 0 t- ϖ 0 T 0 /4). Note that ϖ 0 T 0 /4 = ϖ 0 /4f 0 = 2 π f 0 /4f 0 = π /2 which corresponds to a 90 ° phase producing y(t) = Acos( ϖ 0( t- 90 ° ) or y(t) = Acos( ϖ 0( t- π /2) as illustrated in Figure 1. ° There are other parameters that characterize a sinusoidal signal such a wavelength. This is explored in the next example. Example: A fixed frequency sinusoid signal is assumed to have the from x(t)= A sin( ϖ 0 t+ φ 0 ). Such signals are fundamental to the design and analysis of many modern systems. To illustrate, a sinusoid of the form x(t)= A sin( ϖ 0 t+ φ 0 ) can be used to represent a 1 GHz (L-band) class radio transmission at a frequency ϖ 0 = 2 π f 0 =2 π *10 9 r/s, as well as seismic signal have extremely low frequencies. The radio transceiver problem is motivated in Figure 2. The electromagnetic radio wave propagation velocity (called phase velocity ) is the speed of light, or v p ~3x10 8 m/s. Figure 2: Wireless communication link. What is the period of the signal x(t) in seconds? Figure 1: x(t)=cos( ϖ 0 t) (top) and y(t)=cos( ϖ 0( t - 90 ° ) (bottom). 2 Radio Transmitter Antenna Radio Receiver 999.9375 ~ 1 km T 0 /4
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EEL 3135: Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor The sinusoid’s period T , measured in seconds, is given by T =1/ f 0 =10 -9 =1ns.
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.

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Lesson_3 - EEL 3135 Discrete-Time Signals and Systems Dr Fred J Taylor Professor Lesson Title Sinusoids Lesson Number 03 Introduction Sinusoids are

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