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for all signals we can have laplace transform but we can have fourier transform only for certain cases...Fourier transform needs certain conditions caled as Dirchlet conditions whereas laplace wont need anything like this....
Fourier is a subset of Laplace. Laplace is a more generalized transform.
Fourier is used primarily for steady state signal analysis, while Laplace is used for transient signal analysis. Laplace is good at looking for the response to pulses, step functions, delta functions, while Fourier is good for continuous signals.
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Laplace = analogue signal
Fourier = digital signal
Notes on comparisons between Fourier and Laplace transforms:
The Laplace transform of a function is just like the Fourier transform of the same function, except for two things. The term in the exponential of a Laplace transform is a complex number instead of just an imaginary number and the lower limit of integration doesn't need to start at -∞. The exponential factor has the effect of forcing the signals to converge. That is why the Laplace transform can be applied to a broader class of signals than the Fourier transform, including exponentially growing signals. In a Fourier transform, both the signal in time domain and its spectrum in frequency domain are a one-dimensional, complex function. However, the Laplace transform of the 1D signal is a complex function defined over a two-dimensional complex plane, called the s-plane, spanned by two variables, one for the horizontal real axis and one for the vertical imaginary axis. If this 2D function is evaluated along the imaginary axis, the Laplace transform simply becomes the Fourier transform.
Fourier transform can be discrete time(DTFT) or continous time(CTFT).Laplace transform (LT)can be done only for continous time signals or functions.
CTFT -> s = jw represents the spectrum or frequency content of the signal. But restricts the signal to purely imaginary part
LT -> s= (sigma+jw) expands or enlarges the domain to handle much wider class of signals by giving a real part.
If sigma = 0 then LT = CTFT
Also in LT we have to specify Region Of Convergence(R.O.C) to specify the region in the complex plane where LT exist.