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Unformatted text preview: Chapter 1 Fourier Series & Fourier Transforms 1 Amplitude, Frequency, Phase & Spectrum of a Sinusoid • Consider the following instance of an analog signal , a sinu soidal waveform v ( t ) . This waveform can be described mathematically as v ( t ) = A cos(2 πf t + φ ) (1) where A is the amplitude , f is the frequency , and φ is the phase angle of v ( t ) . • The amplitude A of v ( t ) , is its peak value, and is defined to be positive. The frequency f of v ( t ) is the number of cycles it makes in a second. 1 The duration of a cycle is therefore 1 /f . This quantity is called the period , denoted T . The phase angle φ (specified in radians) represents the fact that the peak is shifted away from the time origin and occurs at time t = − φ/ (2 πf ) . • v ( t ) in (1) can be represented by a pair of complex exponen tials, or phasors , using Euler’s formula : e ± jθ = cos θ ± j sin θ where j = √ − 1 . From Euler’s formula, we have cos( θ ) = 1 2 ( e jθ + e − jθ ); sin( θ ) = 1 2 j ( e jθ − e − jθ ) . Therefore, v ( t ) = A cos( θ z } { 2 πf t + φ ) = A 2 e jφ e j 2 πf t  {z } Phasor 1 + A 2 e − jφ e j 2 π ( − f ) t  {z } Phasor 2 . (2) Thus, v ( t ) can be viewed as the sum of 2 phasors. • Notice that a negative frequency is associated with phasor 2. To understand the concept of negative frequencies, we have the following interpretation of the phasors in (2): 2 The phasors are viewed as rotating vectors in the complex plane. Phasor 1 has length A/ 2 , rotates anticlockwise at a rate of f revolutions per second, and at time t = 0 , makes an angle φ w.r.t. to the positive real axis. Phasor 2 has length A/ 2 , rotates clockwise at a rate of f revolutions per second, and at time t = 0 , makes an angle − φ w.r.t. to the positive real axis. • The following diagram shows the direction of the two phasors as v ( t ) completes one revolution for the case φ = 0 . 3 4 • Observe that each phasor is completely specified by 3 para meters: amplitude, frequency and phase angle....
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 Spring '11
 phoon
 Frequency

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