Physical Meaning of CTFS Coefficients – Spectrum

2.1.3 Physical Meaning of CTFS Coefficients – Spectrum

To understand the physical meaning of spectrum, let us see Fig. 2.5, which shows the major Fourier coefficients c k of a zero-mean rectangular wave for k = −3, −1,

1, and 3 (excluding the DC component c 0 ) and the corresponding time functions

2.1 Continuous-Time Fourier Series (CTFS) of Periodic Signals 71

d 1 cos( ω 0 t+ φ 1 )

Re

d 1 cos( ω 0 t+ φ ) + d cos(3 ω t+ φ ) φ 3

cos d 3 –3 ω 0 3 ω [rad/s]

3 cos(3 ω 0 t+ φ 3 )

0 0 [rad/s]

cos d 1 Fig. 2.5 Physical meaning of complex exponential Fourier series coefficients

+c P 3 The observation of Fig. 2.5 gives us the following interpretations of Fourier

spectrum: Remark 2.2 Physical Meaning of Complex Exponential Fourier Series Coefficients

(1) While the trigonometric or magnitude-and-phase form of Fourier series has only nonnegative frequency components, the complex exponential Fourier series has positive/negative frequency (±kω 0 ) components that are conjugate-symmetric about k = 0, i.e.,

(2.1.6b) P

jφ

(2.1.6c) P

as shown in Sect. 2.1.1. This implies that the magnitude spectrum |c k | is (even) symmetric about the vertical axis k = 0 and the phase spectrum φ k is odd symmetric about the origin.

(2) As illustrated above, the k th component appears as the sum of the positive and negative frequency components

d k e − jφ e − jkω 0 t P

e j kω 0 t

−k e − jkω 0 t (2.1.6) 1 d e jφ k e j kω 0 t

2 + 2 =d k cos(kω 0 t+φ k )

which denote two vectors (phasors) revolving in the opposite direction with positive (counter-clockwise) and negative (clockwise) angular velocities ±kω 0

[rad/s] round the origin, respectively. (3) Figure 2.6 also shows that the spectrum presents the descriptive information of

a signal about its frequency components. To get familiar with Fourier spectrum further, let us see and compare the spec-

tra of the three signals, i.e., a rectangular wave, a triangular wave, and a constant

72 2 Continuous-Time Fourier Analysis

d 1 cos( ω 0 t+ φ 1 )+d 3 cos(3 ω 0 t+ φ 3 )

d 1 Time 0 t

Magnitude d No dc component spectrum

1 cos( ω 0 t+ φ 1 )

fundamental component

td d d 3 d 3 d 3 cos(3 ω 0 t+ φ 3 0 ) 2 F re

d 1 No 2nd harmonic

c 0 y the 3rd harmonic component

Fig. 2.6 Physical meaning of spectrum – time domain vs. frequency domain

(DC: Direct Current) signal depicted in Fig. 2.7. The observations are stated in the following remarks:

Remark 2.3 Effects of Smoothness and Period on Spectrum (1) The smoother a time function is, the larger the relative magnitude of low

frequency components to high frequency ones is. Compare Fig. 2.7(a1–b1), (a2–b2), and (a3–b3).

(cf.) The CTFS of the unit constant function can be obtained from Eq. (E2.1.3) with A = 1 and P = D.

0 k 8 (a1) A rectangular wave with D = 1, P = 2 (b1) Its magnitude spectrum

0 D t 5 –8

0 k 8 (a2) A triangular wave with D = 1, P = 2 (b2) Its magnitude spectrum

0 k 8 (a3) A constant signal

0 t 5 –8

(b3) Its magnitude spectrum ω 0

0 k 8 (a4) A rectangular wave with D = 1, P = 4 (b4) Its magnitude spectrum

Fig. 2.7 The CTFS spectra of rectangular/triangular waves and a DC signal

2.2 Continuous-Time Fourier Transform of Aperiodic Signals 73 (2) The longer the period P is, the lower the fundamental frequency ω 0 = 2π/P

becomes and the denser the CTFS spectrum becomes. Compare Fig. 2.7(a4–b4) with (a1–b1). (cf.) This presages the continuous-time Fourier transform, which will be intro-

duced in the next section. Now, let us see how the horizontal/vertical translations of a time function x(t )

affect the Fourier coefficients. < Effects of Vertical/Horizontal Translations of x(t ) on the Fourier coefficients> Translating x(t ) by ±A (+: upward, −: downward) along the vertical axis causes

only the change of Fourier coefficient d 0 =a 0 for k = 0 (DC component or average value) by ±A. On the other side, translating x(t) along the horizontal (time) axis by

±t 1 (+: rightward, −: leftward) causes only the change of phases (φ k ’s) by ∓kω 0 t 1 , not affecting the magnitudes d k of Eq. (2.1.4b) or |c k | of Eq. (2.1.5b):

′ c (2.1.5b)

x (t − t 1 )e − jkω 0 t

0 1 dt = 1 x (t − t 1 )e − jkω (t−t +t dt

0 t 1 =e 0 − jkω x (t − t 1 )e − jkω (t−t 1 ) (2.1.5b) dt

0 k t e − jkω =c 1 = |c k |∠(φ k − kω 0 t 1 )

(2.1.11) Note that x(t − t 1 ) is obtained by translating x(t ) by t 1 in the positive (rightward)

direction for t 1 > 0 and by −t 1 in the negative (leftward) direction for t 1 < 0 along the horizontal (time) axis. Eq. (2.1.11) implies that horizontal shift of x(t ) causes a change not in its magnitude spectrum but in its phase spectrum.