Fourier Series and Transform

Frequency Analysis

Frequency analysis of any given signal involves the transformation of a time-domain signal into its frequency components. The need for describing a signal in the frequency domain exists because signal processing is generally accomplished using systems that are described in terms of frequency response. Converting the time-domain signals and systems into the frequency domain is extremely helpful in understanding the character-istics of both signals and systems.

In Section 4.1,the Fourier series and Fourier transform will be introduced. The Fourier series is an effective technique for handling periodic functions. It provides a method for expressing a periodic function as the linear combination of sinusoidal functions. The Fourier transform is needed to develop the concept of frequency-domain signal processing. Section 4.2 introduces the z-transform,its important properties,and its inverse transform. Section 4.3 shows the analysis and implementation of digital systems using the z-transform. Basic concepts of discrete Fourier transforms will be introduced in Section 4.4,but detailed treatments will be presented in Chapter 7. The application of frequency analysis techniques using MATLAB to design notch filters and analyze room acoustics will be presented in Section 4.5. Finally,real-time experiments using the TMS320C55x will be presented in Section 4.6.

4.1 Fourier Series and Transform

In this section,we will introduce the representation of analog periodic signals using Fourier series. We will then expand the analysis to the Fourier transform representation of broad classes of finite energy signals.

4.1.1 Fourier Series

Any periodic signal, x(t),can be represented as the sum of an infinite number of harmonically related sinusoids and complex exponentials. The basic mathematical representation of periodic signal x(t) with period T0 (in seconds) is the Fourier series defined as

ISBNs: 0-470-84137-0 (Hardback); 0-470-84534-1 (Electronic)

x…t† ˆ X¹

kˆ 1

c_ke^jkO0t, …4:1:1†

where ckis the Fourier series coefficient,and V0ˆ 2p=T0is the fundamental frequency (in radians per second). The Fourier series describes a periodic signal in terms of infinite sinusoids. The sinusoidal component of frequency kV₀is known as the kth harmonic.

The kth Fourier coefficient, c_k,is expressed as ckˆ 1

T0

…

T0

x…t†e ^jkV0tdt: …4:1:2†

This integral can be evaluated over any interval of length T0. For an odd function,it is easier to integrate from 0 to T0. For an even function,integration from T0=2 to T0=2 is commonly used. The term with k ˆ 0 is referred to as the DC component because c0ˆ_T¹₀„

T0x…t†dt equals the average value of x(t) over one period.

Example 4.1: The waveform of a rectangular pulse train shown in Figure 4.1 is a periodic signal with period T₀,and can be expressed as

x…t† ˆ A, kT⁰ t=2  t  kT0‡ t=2 0,otherwise,



…4:1:3†

where k ˆ 0,  1,  2, . . . ,and t < T₀. Since x(t) is an even signal,it is con-venient to select the integration from T0=2 to T0=2. From (4.1.2),we have

c_kˆ 1 V0 ! 1,and equals 0 at frequencies that are multiples of p. Because the periodic signal x(t) is an even function,the Fourier coefficients ckare real values.

For the rectangular pulse train with a fixed period T0,the effect of decreasing t is to spread the signal power over the frequency range. On the other hand,when t is fixed but the period T₀increases,the spacing between adjacent spectral lines decreases.

t Figure 4.1 Rectangular pulse train

128 FREQUENCY ANALYSIS

A periodic signal has infinite energy and finite power,which is defined by Parseval's

Since cj jk²represents the power of the kth harmonic component of the signal,the total power of the periodic signal is simply the sum of the powers of all harmonics.

The complex-valued Fourier coefficients, c_k,can be expressed as

ckˆ cj jek ^jfk: …4:1:6†

A plot of jckj versus the frequency index k is called the amplitude (magnitude) spectrum, and a plot of f_kversus k is called the phase spectrum. If the periodic signal x(t) is real valued,it is easy to show that c0 is real valued and that ck and c k are complex conjugates. That is,

c_kˆ c^_k, jc _kj ˆ cj j and fk _kˆ f_k: …4:1:7†

Therefore the amplitude spectrum is an even function of frequency V,and the phase spectrum is an odd function of V for a real-valued periodic signal.

If we plot jckj² as a function of the discrete frequencies kV0,we can show that the power of the periodic signal is distributed among the various frequency components.

This plot is called the power density spectrum of the periodic signal x(t). Since the power in a periodic signal exists only at discrete values of frequencies kV0,the signal has a line spectrum. The spacing between two consecutive spectral lines is equal to the funda-mental frequency V₀.

Example 4.2: Consider the output of an ideal oscillator as the perfect sinewave expressed as

x…t† ˆ sin 2pf… 0t†, f0ˆV0

2p:

We can then calculate the Fourier series coefficients using Euler's formula (Appendix A.3) as

This equation indicates that there is no power in any of the harmonic k 6ˆ 1.

Therefore Fourier series analysis is a useful tool for determining the quality (purity) of a sinusoidal signal.

4.1.2 Fourier Transform

We have shown that a periodic signal has a line spectrum and that the spacing between two consecutive spectral lines is equal to the fundamental frequency V0ˆ 2p=T0. The number of frequency components increases as T0is increased,whereas the envelope of the magnitude of the spectral components remains the same. If we increase the period without limit (i.e., T0! 1),the line spacing tends toward 0. The discrete frequency components converge into a continuum of frequency components whose magnitudes have the same shape as the envelope of the discrete spectra. In other words,when the period T₀ approaches infinity,the pulse train shown in Figure 4.1 reduces to a single pulse,which is no longer periodic. Thus the signal becomes non-periodic and its spectrum becomes continuous.

In real applications,most signals such as speech signals are not periodic. Consider the signal that is not periodic (V₀! 0 or T₀! 1),the number of exponential components in (4.1.1) tends toward infinity and the summation becomes integration over the entire continuous range ( 1, 1†. Thus (4.1.1) can be rewritten as

x…t† ˆ 1 2p

…₁

1X…V†e^jVtdV: …4:1:9†

This integral is called the inverse Fourier transform. Similarly,(4.1.2) can be rewritten as

X…V† ˆ …₁

1x…t†e ^jVtdt, …4:1:10†

which is called the Fourier transform (FT) of x(t). Note that the time functions are represented using lowercase letters,and the corresponding frequency functions are denoted by using capital letters. A sufficient condition for a function x(t) that possesses a Fourier transform is

…₁

1jx…t†jdt < 1: …4:1:11†

That is, x(t) is absolutely integrable.

Example 4.3: Calculate the Fourier transform of the function x…t† ˆ e ^atu…t†,where a > 0 and u(t) is the unit step function. From (4.1.10),we have

130 FREQUENCY ANALYSIS

X…V† ˆ …₁

1e ^atu…t†e ^jVtdt

ˆ …₁

0 e ^a‡jV†tdt

ˆ 1

a ‡ jV:

The Fourier transform X…V† is also called the spectrum of the analog signal x(t). The spectrum X…V† is a complex-valued function of frequency V,and can be expressed as

X…V† ˆX…V†e^jf…V†, …4:1:12†

where jX…V†j is the magnitude spectrum of x(t),and f…V† is the phase spectrum of x(t).

In the frequency domain, jX…V†j² reveals the distribution of energy with respect to the frequency and is called the energy density spectrum of the signal. When x(t) is any finite energy signal,its energy is

E_xˆ …₁

1jx…t†j²dt ˆ 1 2p

…₁

1jX…V†j²dV: …4:1:13†

This is called Parseval's theorem for finite energy signals,which expresses the principle of conservation of energy in time and frequency domains.

For a function x(t) defined over a finite interval T0,i.e.,x…t† ˆ 0 for jtj > T0=2,the Fourier series coefficients ckcan be expressed in terms of X…V† using (4.1.2) and (4.1.10) as

ckˆ 1

T₀X kV… 0†: …4:1:14†

For a given finite interval function,its Fourier transform at a set of equally spaced points on the V-axis is specified exactly by the Fourier series coefficients. The distance between adjacent points on the V-axis is 2p=T0 radians.

If x(t) is a real-valued signal,we can show from (4.1.9) and (4.1.10) that

FT x… t†Š ˆ X‰ ^…V† and X… V† ˆ X^…V†: …4:1:15†

It follows that

jX… V†j ˆ jX…V†j and f… V† ˆ f…V†: …4:1:16†

Therefore the amplitude spectrum jX…V†j is an even function of V,and the phase spectrum is an odd function.

If the time signal x(t) is a delta function d…t†,its Fourier transform can be calculated as

X…V† ˆ …₁

1d…t†e ^jVtdt ˆ 1: …4:1:17†

This indicates that the delta function has frequency components at all frequencies. In fact,the narrower the time waveform,the greater the range of frequencies where the signal has significant frequency components.

Some useful functions and their Fourier transforms are summarized in Table 4.1. We may find the Fourier transforms of other functions using the Fourier transform proper-ties listed in Table 4.2.

Table 4.1 Common Fourier transform pairs Time function x…t† Fourier transform X…V†

d…t† 1

d…t t† e ^jVt

1 2pd…V†

e ^atu…t† 1

a ‡ jV

e^jV0t 2pd…V V0†

sin…V0t† jp‰d…V ‡ V0† d…V V0†Š

cos…V0t† p‰d…V ‡ V0† ‡ d…V V0†Š

sgn…t† ˆ 1, t  0 1, t < 0

 2

jV

Table 4.2 Useful properties of the Fourier transform Time function x…t† Property Fourier transform X…V†

a1x1…t† ‡ a2x2…t† Linearity a1X1…V† ‡ a2X2…V†

dx…t†

dt Differentiation in time

domain jVX…V†

tx…t† Differentiation in

frequency domain jdX…V†

dV

x… t† Time reversal X… V†

x…t a† Time shifting e ^jVaX…V†

x…at† Time scaling 1

jajX V a

 

x…t† sin…V0t† Modulation 1

2j‰X…V V0† X…V ‡ V0†Š

x…t† cos…V0t† Modulation 1

2‰X…V ‡ V0† ‡ X…V V0†Š

e ^atx…t† Frequency shifting X…V ‡ a†

132 FREQUENCY ANALYSIS

Example 4.4: Find the Fourier transform of the time function y…t† ˆ e ^ajtj, a > 0:

This equation can be written as

y…t† ˆ x… t† ‡ x…t†, where

x…t† ˆ e ^atu…t†, a > 0:

From Table 4.1,we have X…V† ˆ 1=…a ‡ jV†. From Table 4.2,we have Y…V† ˆ X… V† ‡ X…V†. This results in

Y…V† ˆ 1

a jV‡ 1

a ‡ jVˆ 2a a²‡ V²:

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