Continuous-time signals  (Page 5/5)

The Fourier series expansion results in transforming a periodic, continuous time function, $\tilde{x}\left(t\right)$ , to two discrete indexed frequency functions, $a\left(k\right)$ and $b\left(k\right)$ that are not periodic.

The fourier transform

Many practical problems in signal analysis involve either infinitely long or very long signals where the Fourier series is not appropriate.For these cases, the Fourier transform (FT) and its inverse (IFT) have been developed. This transform has been used with great success invirtually all quantitative areas of science and technology where theconcept of frequency is important. While the Fourier series was used before Fourier worked on it, the Fourier transform seems to be his original idea.It can be derived as an extension of the Fourier series by letting the length or period $T$ increase to infinity or the Fourier transform can be independently defined and then the Fourier series shown to be a special case of it. Thelatter approach is the more general of the two, but the former is more intuitive [link] , [link] .

Definition of the fourier transform

The Fourier transform (FT) of a real-valued (or complex) function of the real-variable $t$ is defined by

giving a complex valued function of the real variable $\omega$ representing frequency. The inverse Fourier transform (IFT) is given by

Because of the infinite limits on both integrals, the question of convergence is important. There are useful practical signals that donot have Fourier transforms if only classical functions are allowed because of problems with convergence. The use of delta functions(distributions) in both the time and frequency domains allows a much larger class of signals to be represented [link] .

Properties of the fourier transform

The properties of the Fourier transform are somewhat parallel to those of the Fourier series and are important in applying it tosignal analysis and interpreting it. The main properties are given here using the notation that the FT of a real valued function $x\left(t\right)$ over all time $t$ is given by $\mathcal{F}\left\{x\right\}=X\left(\omega \right)$ .

1. Linear: $\mathcal{F}\{x+y\}=\mathcal{F}\left\{x\right\}+\mathcal{F}\left\{y\right\}$
2. Even and Oddness: if $x\left(t\right)=u\left(t\right)+jv\left(t\right)$ and $X\left(\omega \right)=A\left(\omega \right)+jB\left(\omega \right)$ then

   $u$	$v$	$A$	$B$	$\left|X\right|$	$\theta$
   even	0	even	0	even	0
   odd	0	0	odd	even	0
   0	even	0	even	even	$\pi /2$
   0	odd	odd	0	even	$\pi /2$

3. Convolution: If continuous convolution is defined by:
   $y\left(t\right)=h\left(t\right)*x\left(t\right)={\int }_{-\infty }^{\infty }h(t-\tau )x\left(\tau \right)d\tau ={\int }_{-\infty }^{\infty }h\left(\lambda \right)x(t-\lambda )d\lambda$
   then $\mathcal{F}\left\{h\right(t)*x(t\left)\right\}=\mathcal{F}\left\{h\right(t\left)\right\}\mathcal{F}\left\{x\right(t\left)\right\}$
4. Multiplication: $\mathcal{F}\left\{h\left(t\right)x\left(t\right)\right\}=\frac{1}{2\pi }\mathcal{F}\left\{h\left(t\right)\right\}*\mathcal{F}\left\{x\left(t\right)\right\}$
5. Parseval: ${\int }_{-\infty }^{\infty }{\left|x\left(t\right)\right|}^{2}dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|X\left(\omega \right)\right|}^{2}d\omega$
6. Shift: $\mathcal{F}\left\{x(t-T)\right\}=X\left(\omega \right)e^{-j\omega T}$
7. Modulate: $\mathcal{F}\left\{x\left(t\right)e^{j2\pi Kt}\right\}=X(\omega -2\pi K)$
8. Derivative: $\mathcal{F}\left\{\frac{dx}{dt}\right\}=j\omega X\left(\omega \right)$
9. Stretch: $\mathcal{F}\left\{x\left(at\right)\right\}=\frac{1}{\left|a\right|}X(\omega /a)$
10. Orthogonality: ${\int }_{-\infty }^{\infty }{e}^{-j{\omega }_{1}t}{e}^{j{\omega }_{2}t}=2\pi \delta ({\omega }_1-{\omega }_2)$

Examples of the fourier transform

Deriving a few basic transforms and using the properties allows a large class of signals to be easily studied. Examples of modulation, sampling,and others will be given.

• If $x\left(t\right)=\delta \left(t\right)$ then $X\left(\omega \right)=1$
• If $x\left(t\right)=1$ then $X\left(\omega \right)=2\pi \delta \left(\omega \right)$
• If $x\left(t\right)$ is an infinite sequence of delta functions spaced $T$ apart, $x\left(t\right)={\sum }_{n=-\infty }^{\infty }\delta (t-nT)$ , its transform is also an infinite sequence of delta functions of weight $2\pi /T$ spaced $2\pi /T$ apart, $X\left(\omega \right)=2\pi {\sum }_{k=-\infty }^{\infty }\delta (\omega -2\pi k/T)$ .
• Other interesting and illustrative examples can be found in [link] , [link] .

Note the Fourier transform takes a function of continuous time into a function of continuous frequency, neither function being periodic. If “distribution" or“delta functions" are allowed, the Fourier transform of a periodic function will be a infinitely long string of delta functions with weights that are the Fourierseries coefficients.

The laplace transform

The Laplace transform can be thought of as a generalization of the Fourier transform in order to include a larger class of functions, to allow theuse of complex variable theory, to solve initial value differential equations, and to give a tool for input-output description of linearsystems. Its use in system and signal analysis became popular in the 1950's and remains as the central tool for much of continuous time systemtheory. The question of convergence becomes still more complicated and depends on complex values of $s$ used in the inverse transform which must be in a “region of convergence" (ROC).

Definition of the laplace transform

The definition of the Laplace transform (LT) of a real valued function defined over all positive time $t$ is

and the inverse transform (ILT) is given by the complex contour integral

where $s=\sigma +j\omega$ is a complex variable and the path of integration for the ILT must be in the region of the $s$ plane where the Laplace transform integral converges. This definition is often called thebilateral Laplace transform to distinguish it from the unilateral transform (ULT) which is defined with zero as the lower limit of the forwardtransform integral [link] . Unless stated otherwise, we will be using the bilateral transform.

Notice that the Laplace transform becomes the Fourier transform on the imaginary axis, for $s=j\omega$ . If the ROC includes the $j\omega$ axis, the Fourier transform exists but if it does not, only the Laplace transform of the function exists.

There is a considerable literature on the Laplace transform and its use in continuous-time system theory. We will develop most of these ideas forthe discrete-time system in terms of the z-transform later in this chapter and will only briefly consider only the more important propertieshere.

The unilateral Laplace transform cannot be used if useful parts of the signal exists for negative time. It does not reduce to the Fouriertransform for signals that exist for negative time, but if the negative time part of a signal can be neglected, the unilateral transform willconverge for a much larger class of function that the bilateral transform will. It also makes the solution of linear, constant coefficient differentialequations with initial conditions much easier.

Properties of the laplace transform

Many of the properties of the Laplace transform are similar to those for Fourier transform [link] , [link] , however, the basis functions for the Laplace transform are not orthogonal. Some of the more important ones are:

1. Linear: $\mathcal{L}\{x+y\}=\mathcal{L}\left\{x\right\}+\mathcal{L}\left\{y\right\}$
2. Convolution: If $y\left(t\right)=h\left(t\right)*x\left(t\right)=\int h(t-\tau )x\left(\tau \right)d\tau$
   then $\mathcal{L}\left\{h\right(t)*x(t\left)\right\}=\mathcal{L}\left\{h\right(t\left)\right\}\mathcal{L}\left\{x\right(t\left)\right\}$
3. Derivative: $\mathcal{L}\left\{\frac{dx}{dt}\right\}=s\mathcal{L}\left\{x\left(t\right)\right\}$
4. Derivative (ULT): $\mathcal{L}\left\{\frac{dx}{dt}\right\}=s\mathcal{L}\left\{x\left(t\right)\right\}-x\left(0\right)$
5. Integral: $\mathcal{L}\{\int x\left(t\right)dt\}=\frac{1}{s}\mathcal{L}\left\{x\left(t\right)\right\}$
6. Shift: $\mathcal{L}\left\{x(t-T)\right\}=C\left(k\right)e^{-Ts}$
7. Modulate: $\mathcal{L}\left\{x\left(t\right)e^{j{\omega }_{0}t}\right\}=X(s-j{\omega }_0)$

Examples can be found in [link] , [link] and are similar to those of the z-transform presented later in these notes. Indeed, note the parallals anddifferences in the Fourier series, Fourier transform, and Z-transform.