Polar Representation and Graphical Plot of CTFT

2.6 Polar Representation and Graphical Plot of CTFT

Noting that a signal x(t ) can be completely recovered from its Fourier trans- form X (ω) via the inverse Fourier transform formula (2.3.1b), we may say that

X (ω) contains all the information in x(t ). In this section we consider the polar or magnitude-phase representation of X (ω) to gain more insight to the (generally complex-valued) Fourier transform. We can write it as

X (ω) = |X(ω)|∠X(ω)

2.6 Polar Representation and Graphical Plot of CTFT 97

(a) φ 1 = φ 2 = φ 3 = 0 [rad]

(b) φ 1 = 3 [rad], φ 2 = 6 [rad], and φ 3 = 9 [rad]

(c) φ 1 = 3 [rad], φ 2 = 2 [rad], and φ 3 = 1 [rad]

Fig. 2.17 Plots of x(t) = 0.5 cos(2πt − φ 1 ) + cos(4πt − φ 2 ) + (2/3) cos(6πt − φ 3 ) with different phases

where |X(ω)| and ∠X(ω) give us the information about the magnitudes and phases of the complex exponentials making up x(t ). Notice that if x(t ) is real, |X(ω)| is an even function of ω and ∠X (ω) is an odd function of ω and thus we need to plot the spectrum for ω ≥ 0 only (see Sect. 2.5.2).

The signals having the same magnitude spectrum may look very different depend- ing on their phase spectra, which is illustrated in Fig. 2.17. Therefore, in some instances phase distortion may be serious.

2.6.1 Linear Phase

There is a particular type of phase characteristic, called linear phase, that the phase shift at frequency ω is a linear function of ω. Specifically, the Fourier transform of x (t ) changed in the phase by −αω, by the time shifting property (2.5.6), simply results in a time-shifted version x(t − α):

X 1 (ω) = X(ω)∠ − αω = X(ω)e − jαω F ↔x 1 (t ) = x(t − α) (2.6.1) For example, Fig. 2.17(b) illustrates how the linear phase shift affects the shape

of x(t ).

2.6.2 Bode Plot

To show a Fourier transform X (ω), we often use a graphical representation consist- ing of the plots of the magnitude |X(ω)| and phase ∠X(ω) as functions of frequency

ω . Although this is useful and will be used extensively in this book, we introduce

98 2 Continuous-Time Fourier Analysis another representation called the Bode plot, which is composed of two graphs,

i.e., magnitude curve of log-magnitude 20 log 10 |X(ω)| [dB] and the phase curve of ∠X (ω) [degree] plotted against the log frequency log 10 ω . Such a representation using the logarithm facilitates the graphical manipulations performed in analyzing LTI systems since the product and division factors in X (ω) become additions and subtractions, respectively. For example, let us consider a physical system whose system or transfer function is

G (1 + T 1 s )(1 + T 2 s (s) = ) (2.6.2) s (1 + T a s )(1 + 2ζ T b s + (T b s ) 2 )

As explained in Sect. 1.2.6, its frequency response, that is the Fourier transform of the impulse response, is obtained by substituting s = jω into G(s):

K (1 + jω T 1 )(1 + jω T 2 )

G ( j ω) = G(s)| s= jω = (2.6.3)

a )(1 + jω2ζ T b − (ωT b ) (1 + jω T 2 ) The magnitude of G( j ω) in decibels is obtained by taking the logarithm on the base

jω

10 and then multiplying by 20 as follows:

|G( jω)| = 20 log 10 |G( jω)|[dB]

= 20 log 10 |K | + 20 log 10 |1 + jω T 1 | + 20 log 10 |1 + jω T 2 |

− 20 log 10 | jω| − 20 log 10 |1 + jω T a |

(2.6.4a) The phase of G( j ω) can be written as ∠G(ω) = ∠K + ∠(1 + jω T 1 ) + ∠(1 + jω T 2 ) − ∠ jω − ∠(1 + jω T a )

− 20 log 10 |1 + jω2ζ T b − (ωT b ) 2 |

(2.6.4b) The convenience of analyzing the effect of each factor on the frequency response

− ∠(1 + jω2ζ T 2

b − (ωT b ) )

explains why Bode plots are widely used in the analysis and design of linear time- invariant (LTI) systems.

(cf.) The MATLAB function “ bode(n,d,..)” can be used to plot Bode plots.