ELEC 36 LECURE NOES WEEK 9: Chapters 7&9 Chapter 7 (cont d) Discrete-ime Processing of Continuous-ime Signals It is often advantageous to convert a continuous-time signal into a discrete-time signal so that processing is done in the discretetime domain using a discrete-time processor such as a computer or a microcomputer. Nevertheless, after processing, it is always the case that the discrete-time signal is converted back into a continuous-time signal. he recovered continuous-time signal could be similar to the original signal, or an altered version of it, depending on the application. As you may guess, the above process consists of three parts: () sampling, (2) discrete-time processing, and (3) converting the processed signal into a continuous-time signal. his three-step process in illustrated in the following figure. (t) x c conversion to D x d [n] discrete-time system y d [n] conversion to C y c (t) Fig. 9.. D processing of C signals. Form the above figure, we can see that the output of the sampler is x c (n ) where is the sampling period, and n is integer. For convenience, we represent x c (n ) by x d [n] x c (n ) so mathematically we can say that x d [n] is a discrete-time signal. he output of the D system is also discrete, denoted by y d [n]. y d [n] is then converted back into a C signal, denoted by y c (t). he conversion of x c (t) into x d [n] is called continuous-to-discrete (D/C), and the conversion of y d [n] into y c (t) is called discrete-tocontinuous (C/D).
We elaborate further on the relationship between the signals above in the following block diagram. (t) x c C/D conversion xd [ n] xc ( n ) discrete-time system yd [ n] yc ( n ) D/C conversion y c (t) Fig. 9.2. Notation for C/D conversion and D/C conversion. Note if we consider the sampled sequence x c (n ), the spacing between adjacent samples is, whereas the spacing between adjacent samples in the sequence x d [n] is unity although both sequences correspond to the same signal. his is because x c (n ) is plotted against time (the x-axis), whereas x d [n] is plotted against n, which is an integer and the spacing between two consecutive integers is. his is illustrated in the following figure. C/D conversion x(t) p(t) x p (t) Conversion of impulse train to discrete-time sequence x d [n] x(t) t 2 x p (t) x p (t) -3-2 - 2 3 t - x d [n] x d [n] -3-2 - 2 3 n - Fig. 9.3. Sampling with a periodic impulse train followed by conversion to a discrete-time sequence. 2 n
Now we examine the processing stages described above in the frequency domain. Let X p (jw) be the Fourier transform (F) of x p (t), which can be expressed in terms of the sample values of x c (t) as x p (t) n x c (n )δ(t n ) (9.) which is simply an impulse train except that the n th impulse is weighted by x c (n ). Recall that the F of an impulse train is an impulse train, and in this case it is X p (jw) n x c (n )e jwnt (9.2) where this follows from the fact that the F of δ(t n ) is e jwnt. We now consider the F of x d [n] which is given by or equivalently, X d (e jω ) X d (e jω ) n n x d [n]e jωn (9.3) x c (n )e jωn. (9.4) By comparing equations (9.2) and (9.4), we observe that X p (jw) and X d (e jω ) are related through X d (e jω )X p (j Ω ). We also know that X p (jw) simply consists of an infinite number of replicas of X c (jw) centered an integer multiple of w s, i.e., Consequently, X p (jw) X d (e jω ) n n X c (j(w nw s )). X c (j( Ω 2πn )). he relationship between X c (jw), X p (jw) and X d (e jω ) is illustrated in the following figure. 3
X ( jw) w M wm w X p ( jw) X p ( jw) 2 2 2 2 π jω X d ( e ) 2π w 2π 2π w 2 2 jω X d ( e ) 2 2 2 2π 2π Ω 2π 2π Ω Fig. 9.4. Relationship between X c (jw), X p (jw), and X d (e jω ) for different sampling rates. 4
Chapter Nine he Laplace ransform It was mentioned in an earlier chapter that the response of an LI system with impulse response h(t) to a complex exponential input of the form e st is y(t) H(s)e st where H(s) R h(t)e st dt. (9.5) If we let s jw (pure imaginary), the integral in (9.5) is essentially the Fourier transform of h(t). For arbitrary values of the complex variable s, this expression is referred to as the Laplace transform of h(t). herefore, the Laplace transform of a general signal x(t) is defined as X(s) R x(t)e st dt. (9.6) Note that s isacomplexvariable,whichcanbeexpressedingeneral as s σ + jw. when s jw (9.6) becomes X(jw) R x(t)e jwt dt which is the Fourier transform of x(t). herefore, the Fourier transform is a special case of the Laplace transform. Equation (9.6) can also be expressed as X(σ + jw) R R x(t)e (σ+jw)t dt x(t)e σt e jwt dt which is essentially the Fourier transform of the signal x(t)e σt. Example: Let x(t) e at u(t) 5
he Fourier transform X(jw), with a>, is X(jw) R R e at u(t)e jwt dt e at e jwt dt jw + a On the other hand, the Laplace transform of x(t) is or X(s) R X(σ + jw) R R e at u(t)e st dt e (s+a)t dt e (a+σ)t e jwt dt jw + a + σ Since s σ + jw, the last equation becomes Conclusion: X(s) s + a e at u(t) L s + a, where a + σ> Re {s} > a. Re {s} > a. We conclude from the above example that the Laplace transform exists for this particular x(t) only if Re {s} > a. he region in the complex plane in which the Laplace transform exists (or converges) is called region of convergence (ROC). he ROC for the above example is given in the following figure. Im ROC Re -a 6
Example: Let hen x(t) e at u( t) X(s) R R s + a e at u( t)e st dt e at e st dt which converges if Re {s + a} < Re {s} < a, which is illustrated below. Im ROC Re Example: Let x(t) 3e 2t u(t) 2e t u(t) Applying the Laplace transform to x(t) yields X(s) R 3 R -a 3e 2t u(t) 2e t u(t) e st dt e 2t u(t)e st dt 2 R 3 s +2 2 s + e t u(t)e st dt where for these integrals to converge we must have Re {s} > 2 for the first term and Re {s} > for the second term. herefore, the ROC is the intersection of the ROCs for the individual terms, i.e., the overall ROC is Re {s} >. 7
Example: Let X(s) x(t) e 2t u(t)+e t cos (3t) u(t) e 2t + 2 e ( 3j)t + with the condition that 2 e (+3j)t u(t) s +2 + µ + µ 2 s +( 3j) 2 s +(+3j) 2s 2 +5s +2 (s 2 +2s +)(s +2) Re {s} > 2, for the first term and Re {s} > for the second term. herefore, the ROC is the region where Re {s} >, which is the intersection of the individual regions. Example: Let x(t) δ(t) X(s) and the ROC is the entire s plane. 8
he Region of Convergence for Laplace ransform Let X(s) be the Laplace transform of some signal x(t). he ROC of X(s), in general, has the following characteristics:. he ROC of X(s) consists of strips parallel to the jw-axis in the s-plane. 2. For rational Laplace transforms, the ROC doesn t contain any poles. 3. If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. 4. If x(t) is right-sided, and if the line Re {s} σ is in the ROC, then all values of s for which Re {s} >σ will also be in the ROC. 5. If x(t) is left-sided, and if the line Re {s} σ is in the ROC, then all values of s for which Re {s} <σ will also be in the ROC. 6. If x(t) is two sided, and if the line Re {s} σ is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line Re {s} σ. 7. If the Laplace transform X(s) of x(t) is rational, then the ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC. 8. If the Laplace transform X(s) of x(t) is rational, then if x(t) is rightsided, the ROC is the region in the s-plane to the right of the rightmost pole. If x(t) is left sided, the ROC is the region in the s-plane to the left of the leftmost pole. Example: Let X(s) (s +)(s +2) Clearly, there are two poles: s and s 2. his yields three possibilities for the ROC where each possibility corresponds to a different signal. hese possibilities are:. Re {s} > he signal must be right-sided. 2. Re {s} < 2 he signal must be left-sided. 3. 2 < Re {s} < he signal must be two-sided. 9