Relation between fourier laplace and z transform pairs
Including the effects of aliasing, the time domain signal is given by: To eliminate the effects of aliasing from this equation, imagine that the frequency domain is so finely sampled that it turns into a continuous curve.
This makes the time domain infinitely long with no periodicity. The DTFT is the Fourier transform to use here, resulting in the time domain signal being given by the relation: This equation is very important in DSP, because the rectangular pulse in the frequency domain is the perfect low-pass filter.
Therefore, the sinc function described by this equation is the filter kernel for the perfect low-pass filter. This is the basis for a very useful class of digital filters called the windowed-sinc filters, described in Chapter This transform pair isn't as important as the reason it is true.
A 2M - 1 point triangle in the time domain can be formed by convolving an M point rectangular pulse with itself. Since convolution in the time domain results in multiplication in the frequency domain, convolving a waveform with itself will square the frequency spectrum.
Is there a waveform that is its own Fourier Transform? The answer is yes, and there is only one: the Gaussian. Figure e shows a Gaussian curve, and f shows the corresponding frequency spectrum, also a Gaussian curve. This relationship is only true if you ignore aliasing. A similar relationship exists between the laplace transform and the fourier transform of a continuous time signal. Example 1. As shown in example 1 when a signal is sampled in the time domain its laplace transform, and hence, the s-plane, becomes periodic with respect to the j-axis.
Example 2. Determine the z-transform, the region of convergence and the fourier transform of the following signal.
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Relation between fourier laplace and z transform pairs fandango special events
Relationship of Z Transform with Laplace Transform and Fourier Transform.
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I found two excellent resources. This smoothie analogy for the Fourier transforms by Kalid and this overview of transforms by Abdel Helim Zekry. This is my interpretation of what these signal transformations mean in the context of DSP and why we need them. Note that the scope of these transforms is far more wide-reaching than just DSP.
This is just a crude attempt at creating a skyscraper look into the transforms. So here we go. The analogy In some recipes of making a cheesecake, we start with solid cream cheese. Melt it into the cream batter while making the cake. And then cool it to a more solid form before we eat it. The accuracy of this recipe is not essential. But the important takeaway is that we start with one form of the cheese: cold, semi-solid. Melt it to make sure the ingredients mix up well.
And then bring it back to the cold semi-solid form before eating it. The important takeaway is that the cheese needs to be converted into a separate form for us to enhance its taste. Similarly, the signal transforms that we use in DSP have the same purpose.
The s-planes, the z-planes are used to contain an alternate form of the signals. Signals , as we know, exist as a function of time. And to analyze and modify these signals in the physical world, we need to have some mathematical way of representing these equations.
However, in the time domain, we cannot perform certain operations. For example, if you are listening to music, and you want to increase the bass or some audio component that lies in a specific frequency range. It will be much easier if we have the signal in the form of its frequencies.
Then we can select the range of frequency we want to boost and do some operation on just that part. This should now be giving you an idea of why we need to transform signals into different forms. Similar to the frequency domain, the Laplace transform defines a new domain or plane. The s-plane. The importance of the s-plane is that it allows us to convert the differential equation of the time domain signal into an algebraic equation in the s-domain.
Algebra is easier than calculus for us, as well as for machines. Thus, transforming the signal using the Laplace transform makes it easier to perform certain operations on it. The smoothie analogy by Kalid defines the Fourier transform in a similar way. His approach defines the Fourier transform as something that filters out all the components of a smoothie.
Each strip maps onto a different Riemann surface of the z "plane". Mapping of different areas of the s plane onto the Z plane is shown below. IJSER Summary Transforms are used because the time-domain mathematical models of systems are generally complex differential equations.
Transforming these complex differential equations into simpler algebraic expressions makes them much easier to solve. Once the solution to the algebraic expression is found, the inverse transform will give you the time-domain response.
Laplace is used for stability studies and Fourier is used for sinusoidal responses of systems. 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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