ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the

Transcription

1 he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus time t, amplitude versus angular frequency, or amplitude versus frequency f. Figure. (a) Amplitude-time plot, and (b) amplitude-angular frequency plot. = π/ is called the fundamental angular frequency and = is called the second harmonic of the fundamental. In general, n = n is said to be the nth harmonic of the fundamental, where n >. In communication engineering we are interested in steady-state analysis much of the time. he Fourier series provides a useful model for analysing the frequency content and the steady-state network response for periodic input signals. rigonometric (Quadrature) Fourier Series [] A periodic time function s(t) over the interval a - represented by an infinite sum of sinusoidal waveforms < t < a + may be a + (a n cos n t + b n sin n t) (.) n = where is the period of the fundamental frequency f and f = /. his is called the trigonometric (quadrature) Fourier series representation of the time function s(t). he coefficients a n and b n are given by a + / a n = s(t) cos n t dt, n > (.) a / and.

2 a + / b n = s(t) sin n t dt, n > (.3) a / he choice of a is arbitrary, and it is usually set to. Many forms of the trigonometric Fourier series may be written. For example, a' + (a' n cos n t + b' n sin n t) (.4) n = is commonly used. he coefficients a' n and b' n are given by a' n = a + / s(t) cos n t dt, n > (.5) a / and b' n = a + / s(t) sin n t dt, n > (.6) a / Example. Find the trigonometric Fourier series for the periodic time function s(t) shown in Figure.. / Figure. A periodic rectangular waveform. a n = s(t) cos n t dt / τ / a n = A m cos n t dt τ / sin n τ / sin n τ / a n = A m n = A m τ n τ /.

3 b n = / s(t) sin n t dt = / herefore, A m τ + n = sin n τ / (A m τ ) cos n n τ / t. Exponential (Complex or Phasor) Fourier Series [] he time function s(t) may be represented over the interval a - the equivalent exponential (complex or phasor) Fourier series < t < a + by c n e jn t (.7) n = where the coefficients c n are given by a + / c n = s(t) e -jn t dt (.8) a / c is equivalent to the dc value of the waveform s(t). c n is, in general, a complex number. Furthermore, it is a phasor since it is the coefficient of e jn t. he complex Fourier series is easier to use for analytical problems. Many forms of the complex Fourier series may be written. For example, c' n e jn t (.9) n = is commonly used. he coefficients c' n are given by c' n = a + / s(t) e -jn t dt (.) a /.3

4 Example. Find the complex Fourier series for the periodic time function s(t) shown in Figure.. / c n = s(t) e -jn t dt / τ / c n = A m e -jn t dt τ / c n = A ejn τ / e jn τ / m jn sin n τ / sin n τ / c n = A m n = A m τ n τ / herefore, n = sin n τ / (A m τ )e jn t. n τ / he frequency spectrum is shown in Figure.3. Figure.3 Frequency spectrum of a periodic rectangular waveform. Figure.4 shows the effect on the frequency spectrum of smaller τ []. Figure.4 Effect on frequency spectrum of smaller τ. If the bandwidth B is specified as the width of the frequency band of a waveform from zero frequency to the first zero crossing, then B = /τ Hz. If we let the pulse width τ in Figure.4 go to zero and the amplitude A m go to infinity with A m τ =, all spectral lines in the frequency domain have unity length. Figure.5 shows the periodic unit impulses and the frequency spectrum of the periodic unit impulses []. he bandwidth becomes infinite. Figure.5 (a) Periodic unit impulses, and (b) frequency spectrum..4

5 Properties of the Complex Fourier Series [3]. If s(t) is real, c n = c* -n (.). If s(t) is real and even, s(-t), Img[c n ] = (.) 3. If s(t) is real and odd, - s(-t), Re[c n ] = (.3) 4. he complex Fourier-series coefficients of a real waveform are related to the quadrature Fourier-series coefficients by [] c n = a n jb n, n > a, n= a n + jb n, n < (.4) c n = a n + bn (.5) represents the amplitude spectrum and c n = θ n = tan - b n a n (.6) represents the phase spectrum of the real waveform. Proof. a + (a n cos n t + b n sin n t) n = a + [(a n - jb n )e j n t + (a n + jb n )e -j n t ] n =.5

6 a + [c n e j n t + c* n e -j n t ] n = a + [c n e j n t + c -n e -j n t ] n = a + ( c n e j n t ) + n = (c - e -j t + c - e -j t +...) a + ( c n e j n t ) + ( cn e n = j n t ) n = n = c n e jn t herefore, c n = a n + bn. Also, we can write a + a n +b n a ( n n = a cos n t + n +bn a + c n cos (n t + θ n ) n = where c n = θ n = tan - b n a. n b n a sin n t) n +bn he equivalence between the Fourier series coefficients is demonstrated in Figure.6. Figure.6 Fourier series coefficients, n >. Parseval s heorem for the Fourier Series [, 4] Parseval s heorem for the Fourier series states that, if s(t) is a periodic signal with period, then the average normalised power (across a Ω resistor) of s(t) is.6

7 P = a + / s(t) dt = c n (.7) a / n = If s(t) is real, s(t) is simply replaced by s(t). References [] M. Schwartz, Information ransmission, Modulation, and Noise, 4/e, McGraw- Hill, 99. [] P. H. Young, Electronic Communication echniques, 4/e, Prentice-Hall, 998. [3] L. W. Couch II, Digital and Analog Communication Systems, 5/e, Prentice Hall, 997. [4] H. P. Hsu, Analog and Digital Communications, McGraw-Hill,

8 3 A A -A -3 A s ( t ) 3 A sin t A sin t = π f = π (a) ime 3 A A -A -3 A s ( t ) Fundamental = nd harmonic = n = n, n > (b) Figure. (a) Amplitude-time plot, and (b) amplitude-angular frequency plot. s( t ) Am ime - τ τ - - Figure. A periodic rectangular waveform. c n Am τ Envelope π First zero crossing π - π 4π 6π τ τ τ τ Figure.3 Frequency spectrum of a periodic rectangular waveform..8

9 s( t ) Am - s ( t ) A m - s ( t ) A m τ = /4 ime τ= /8 ime τ = /3 c n Am τ Envelope π First zero crossing π 4π 6π τ τ τ c n Am τ c n Am τ π τ π ( π τ π 4π τ at 3nd harmonics) - ime Figure.4 Effect on frequency spectrum of smaller τ. δ ( t + δ ( t ) δ ( t - c ) ) n ime - - π π 4π - 4π (a) (b) Figure.5 (a) Periodic unit impulses, and (b) frequency spectrum..9

10 Imaginary θ n = c n a n Real c n - b n Figure.6 Fourier series coefficients, n >..