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- 1. Z-Transforms<br />AakankshaThakre<br />AyushAgrawal<br />KunalAgrawal<br />AkshayPhadnis<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 2. Introduction:<br />Just like Laplace transforms are used for evaluation of continuous functions, Z-transforms can be used for evaluating discrete functions.<br />Z-Transforms are highly expedient in discrete analysis ,<br />Which form the basis of communication technology.<br />Definition:<br />If a function f(n) is defined for discreet values ( n=0,+1 or -1 , +2 or -2,etc ) & f(n)=0 for n<0,then z-transform of the function is defined as<br />Z{f(n)}=<br />∞<br />∑<br />-n<br />f(n) z<br />=F(z)<br />n=0<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 3. Some standard results & formulae:<br /> -n<br />∞<br /> n=0<br />2<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 4. Z{<br />}<br />n<br />=<br />a<br />Z / (z-a)<br />Z{<br />n<br />}<br />=<br />-a<br />Z / (z+a)<br />2<br />Z{n}= z / <br />(z-1)<br />Z{1/n!}=<br />e<br />1/z<br />Z{sin nф}= zSinф / (z -2zcosф +1 )<br />2<br />2 <br />2<br />Z{Cosnф}= z- zCosф / (z -2zCosф + 1)<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 5. Properties:<br />Linearity: -<br />Z{a f(n)+b g(n)}=a Z{f(n)}+b Z{g(n)}<br />Damping rule:-<br />Z{a f(n)} = F(z/a)<br />Multiplication by positive integer n :-<br />Z{n f(n)}= -z d/dz ( F(z) )<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 6. Initial value theorem:-<br />f(0)= lim F(z)<br />Z∞<br />Final value theorem:-<br />f(∞)= lim f(n) = lim (z-1) F(z)<br />n∞ <br />Z1<br />Shifting Theorem:-<br />Z{ f (n+k) }= z [ F(z) - ∑ f(i) z ]<br />K-i<br />-i<br />K<br />i=0<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 7. Division by n property:-<br />∞<br />∫<br />Z{f(n)/n}=<br />F(z)/z<br />dz<br />Z<br />Division by n+k property:-<br />∞<br />∫<br />k<br />Z{f(n) /(n+k)}=<br />Z<br />K+1<br />F(z)/ (Z)<br />dz<br />z<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 8. Applications of Z-Transforms<br />The field of signal processing is essentially a field of signal analysis in which they are reduced to their mathematical components and evaluated. One important concept in signal processing is that of the Z-Transform, which converts unwieldy sequences into forms that can be easily dealt with. Z-Transforms are used in many signal processing systems<br />Z-transforms can be used to solve differential equations with constant coefficients.<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 9. Derivation of the z-Transform<br />The z-transform is the discrete-time counterpart of the Laplace transform. In this<br />section we derive the z-transform from the Laplace transform a discrete-time signal.<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 10. The Laplace transform X(s), of a continuous-time signal x(t), is given by the integral <br />∞ -st<br />X(s) = ∫ x(t) e dt<br />0-<br />where the complex variable s=a +jω, and the lower limit of t=0− allows the possibility<br />that the signal x(t) may include an impulse. The inverse Laplace transform is defined<br />by:-<br />a+j∞<br />st<br />X(t) = ∫ X(s) e ds<br />a-j∞<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 11. where a is selected so that X(s) is analytic (no singularities) for s>a. The ztransform<br />can be derived from Eq. by sampling the continuous-time input signal<br />x(t). For a sampled signal x(mTs), normally denoted as x(m) assuming the sampling<br />period Ts=1, the Laplace transform Eq. becomes<br />∞<br />s<br />-sm<br />X(e ) = ∑ x(m) e<br />m=0<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 12. Substituting the variable e to the power s <br />in Eq. with the<br />variable z we obtain the one-sided <br />ztransform<br />∞<br />-m<br />X(z) = ∑ x(m) z<br />m = 0<br />The two-sided z-transform is defined as:-<br />∞<br />-m<br />X(z) = ∑ x(m) z<br />m = -∞<br />Aakanksha_Kunal_Ayush_Akshay<br />
- 13. The Relationship Between the Laplace, the Fourier, andthe z-Transforms :-The Laplace transform, the Fourier transform and the z-transform are closely related inthat they all employ complex exponential as their basis function. For right-sidedsignals (zero-valued for negative time index) the Laplace transform is a generalisation of the Fourier transform of a continuous-time signal, and the z-transform is ageneralisation of the Fourier transform of a discrete-time signal.<br />Aakanksha_Kunal_Ayush_Akshay<br />

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