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Difference Between Laplace Transform And Z-Transform


If you are Searching for the difference between Laplace and z transform then here are the Relationship and Difference Between The Z-Transform And The Laplace Transform.








Z-transform

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of
z =def esT 
where T=1/fs is the sampling period (in units of time e.g., seconds) and fs is
 the sampling rate (in samples per second or hertz)



Let

ΔT(t) =def n=0δ(tnT)
be a sampling impulse train (also called a Dirac comb) and
xq(t)=def x(t)ΔT(t)=x(t)n=0δ(tnT)=n=0x(nT)δ(tnT)=n=0x[n]δ(tnT)
be the continuous-time representation of the sampled x(t) \
x[n] =def x(nT) 
are the discrete samples of x(t) The Laplace transform of the sampled signal x_q(t) \ is
Xq(s)=0xq(t)estdt=0n=0x[n]δ(tnT)estdt=n=0x[n]0δ(tnT)estdt=n=0x[n]ensT.
This is precisely the definition of the unilateral Z-transform of the discrete function x[n] .
X(z)=n=0x[n]zn
with the substitution of zesT .
Comparing the last two equations, we find the relationship between the unilateral Z-transform
 and the Laplace transform of the sampled signal:
Xq(s)=X(z)|z=esT
The similarity between the Z and Laplace transforms is expanded upon in the theory of



Laplace Transform


The main drawback of fourier transform (i.e. continuous F.T.) is that it can be defined only for stable systems. Where as, Laplace Transform can be defined for both stable and unstable systems.
Following are the Laplace transform and inverse Laplace transform equations.
Laplace Transform
Following table mentions Laplace transform of various functions.
Laplace Transform of functions
To convert Laplace transform to Fourier tranform, replace s with j*w, where w is the radial frequency. in units of radians per second (rad/s).
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