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Chapter 2 Frequency Domain Analysis of Signals and Systems. CONTENTS. Fourier Series Fourier Transforms Power and Energy Sampling of Bandlimited Signals Bandpass Signals. 2.1 FOURIER SERIES.
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CONTENTS • Fourier Series • Fourier Transforms • Power and Energy • Sampling of Bandlimited Signals • Bandpass Signals
2.1 FOURIER SERIES • Theorem 2.1.1 [Fourier Series] Let the signal x(t) be a periodic signal with period T0.If the following conditions are satisfied 1. x(t) is absolutely integrable over its period 2. The number of maxima and minima of x(t) in each period is finite 3. The number of discontinuous of x(t) in each period is finite then x(t) can be expanded in terms of the complex exponential signal as where for some arbitrary and
xnare called the Fourier series coefficients of the signal x(t). • For all practice purpose, • From now on, we will use instead of • The quantity is called the fundamental frequency of the signal x(t) • The Fourier series expansion can be expressed in terms of angular frequency by and
Discrete spectrum - we may represent where gives the magnitude of the nth harmonic and gives its phase.
Example: Let x(t) denote the periodic signal depicted in Figure 2.2 where is a rectangular pulse. Determine the Fourier series expansion for this signal
2.2.1 Fourier Series for Real Signals • If the signal x(t) is a real signal, we have
For a real, periodic x(t), the positive and negative coefficients are conjugates. • has even symmetry and has odd symmetry with respect to the n=0 axis.
Since is real and , we conclude that • We may let and • This relation is called the trigonometric Fourier series expansion.
To obtain and , we have • From above equation, we obtain
There exists a third way to represent the Fourier series expansion of a real signal. Noting that we have • For a real periodic signal, we have three alternative ways to represent Fourier series expansion
2.2 FOURIER TRANSFORMS • Fourier transform is the extension of Fourier series to periodic and nonperiodic signals. • The signal are expressed in terms of complex exponentials of various frequencies, but these frequencies are not discrete. • The signal has a continuous spectrum as opposed to a discrete spectrum.
Theorem 2.2.1 [Fourier Transform] If the signal x(t) satisfies certain conditions known as Dirichlet conditions, namely, 1. x(t) is absolutely integrable on the real line, i.e., 2. The number of maxima and minima of x(t) in any finite interval on the real line is finite, 3. The number of discontinuities of x(t) in any finite interval on the real line is finite, Then, the Fourier transform of x(t), defined by And the original signal can be obtained from its Fourier transform by
Observations • X(f) is in general a complex function. The function X(f) is sometimes referred to as the spectrum of the signal x(t). • To denote that X(f) is the Fourier transform of x(t), the following notation is frequently employed to denote that x(t) is the inverse Fourier transform of X(f) , the following notation is used Sometimes the following notation is used as a shorthand for both relations
The Fourier transform and the inverse Fourier transform relations can be written as On the other hand, where is the unit impulse. From above equation, we may have or, in general hence, the spectrum of is equal to unity over all frequencies.
Example 2.2.1: Determine the Fourier transform of the signal . Solution: We have
Example 2.2.2: Find the Fourier transform of the impulse signal . Solution: The Fourier transform can be obtained by Similarly, from the relation We conclude that
2.2.1 Fourier Transform of Real, Even, and Odd Signals • The Fourier transform can be written in general as For real x(t),
Since cosine is an even function and sine is an odd function, we see that, for real x(t), the real part of X(f) is an even function of f and the imaginary part is an odd function of f. Therefore, we have • This is equivalent to the following relations:
2.2.2 Basic Properties of the Fourier Transform • Linearity Property: Given signals and with the Fourier transforms The Fourier transform of is
Duality Property: • Time Shift Property: A shift of in the time origin causes a phase shift of in the frequency domain. • Scaling Property: For any real , we have
Convolution Property: If the signal and both possess Fourier transforms, then • Modulation Property: The Fourier transform of is , and the Fourier transform of is
Parseval’s Property: If the Fourier transforms of and are denoted by and , respectively, then • Rayleigh’s Property: If X(f) is the Fourier transform of x(t), then
Autocorrelation Property: The (time) autocorrelation function of the signal x(t) is denoted by and is defined by The autocorrelation property states that • Differentiation Property: The Fourier transform of the derivative of a signal can be obtained from the relation
Integration Property: The Fourier transform of the integral of a signal can be determined from the relation • Moments Property: If , then , the nth moment of x(t), can be obtained from the relation
2.2.3 Fourier Transform for Periodic Signals • Let x(t) be a periodic signal with period , satisfying the Dirichlet conditions. Let denote the Fourier series coefficients corresponding to this signal. Then • Since we obtain
If we define the truncated signal as we may have
By using the convolution theorem, we obtain Comparing this result with we conclude
Alternative way to find the Fourier series coefficients, Given the periodic signal x(t), we carry out the following steps to find : 1. Find the truncated signal . 2. Determine the Fourier transform of the truncated signal. 3. Evaluate the Fourier transform of the truncated signal at , to obtain the nth harmonic and multiply by
Example 2.2.3: Determine the Fourier series coefficients of the signal x(t) shown in Figure 2.2. Solution: The truncated signal is and its Fourier transform is Therefore,
2.3 POWER AND ENERGY • The energy content of a signal x(t), denoted by , is defined as and the power content of a signal is • A signal is energy-type if and is power-type if • A signal cannot be both power- and energy-type because for energy-type signals and for power-type signals
All nonzero periodic signals with period are power-type and have power where is any arbitrary real number.
2.3.1 Energy-Type Signal • For any energy-type signal x(t), we define the autocorrelation function as • By setting , we obtain
This relation gives two methods for finding the energy in a signal. One method uses x(t), the time-domain representation of the signal, and the other method uses X(f) , the frequency-domain representation of the signal. • The energy spectral densityof the signal x(t) is defined by • The energy spectrum density represents the amount of energy per hertz of bandwidth present in the signal at various frequencies.
2.3.2 Power-Type Signals • Define the time-average autocorrelation function of the power-type signal x(t) as • The power content of the signal can be obtained from
Define , the power-spectral density or the power spectrum of the signal x(t) to be the Fourier transform of the time-average autocorrelation function • Now we may express the power content of the signal x(t) in terms of , i.e.,
If a power-type signal x(t) is passed through a filter with impulse response h(t), the output is The time-average autocorrelation function for the output signal is
By making a change of variables and changing the order of integration we obtain where in (a) we have use the definition of and in (b) and (c) we have used the definition of the convolution integral.
Taking the Fourier transform of both sides of above equation, we obtain
For periodic signals, the time-average autocorrelation function and the power spectral density can be simplified considerably. Assume that x(t) is a periodic signal with period having the Fourier series coefficients . The time-average autocorrelation function can be expressed as integrated over one period
If we substitute the Fourier series expansion of the periodic signal in this relation, we obtain • Now using the fact that we obtain • Time-average autocorrelation function of a periodic signal is itself periodic with the same period as the original signal, and its Fourier series coefficients are magnitude squares of the Fourier series coefficients of the original signal.
The power spectral density of a periodic signal • The power content of a periodic signal This relation is known as Rayleigh’s relation for periodic signals.