Laplace Transform an intuitive approach

From the inverse Fourier

transform

integral, any signal in the time domain can be decomposed into a linear combination of complex exponentials of all frequencies from – infinity to + infinity. For real-valued functions we can treat these complex exponentials as pairs that have positive and negative frequencies. Now let's remember Euler's formula when we add 2 complex exponentials with opposite frequencies, the imaginary part vanishes and we are left with a real sinusoid with double amplitude and oscillating with the same frequency as its parents. Therefore, any real-valued signal in the time domain can be decomposed into a linear combination of sinusoids. You can revisit the Fourier series and Fourier

transform

videos here on this channel for more information.

But the bottom line here is that if we track the amplitude of these sinusoids as well as the amount of shift from the zero position at all frequencies, we get the amplitude and phase spectrum of a signal. Each point on these graphs hides a sinusoid that oscillates with a certain frequency, which has a certain amplitude and a certain amount of phase shift. That's all about the Fourier transform. How about Laplace? If we look at the Laplace transform formula. It looks exactly the same as Fourier but instead of j omega there is S in the exponential function.

Now S is a complex quantity consisting of real and imaginary parts. By expanding S, you can easily see that the Laplace transform is the Fourier transform multiplied by a factor (e raised to the minus sigma t). This factor is known as the convergence factor because it sometimes helps the Fourier transform exist. The existence here means that the integral of the Fourier transform converges. Sigma can take any value from minus infinity to plus infinity as long as the Laplace integral converges. Replacing sigma with zero, the Laplace transform is equal to the Fourier transform. For example, the Fourier transform of the rectangular pulse has the form of the sinc function.

When sigma is equal to zero, the Laplace transform is the same as the Fourier transform. Now, if we find the allowed values ​​of sigma and take the Fourier transform at each value of sigma, we get the full Laplace transform of the function. Just like the previous Laplace transform has its own magnitude and phase spectrum, but now because Laplace has two degrees of freedom sigma and j omega, the magnitude and phase spectrum are 3D surfaces instead of normal 2D graphs. I want you to see clearly that the Fourier transform is only a portion of the Laplace transform.

Normally it is the Laplace transform when the value of sigma is equal to zero. Therefore, the Fourier magnitude and phase spectrum are plotted against the frequency axis, while the Laplace ones are plotted in what is called the s-plane. The sigma values ​​in the S plain where the Laplace transform exists determine the region of convergence (ROC). If the region of convergence contains the j-omega axis, then the Fourier transform exists. So, there are functions whose Fourier transform does not exist, while the Laplace transform exists for restricted values ​​of sigma. Note that the omega j axis is the frequency axis in the Fourier transform.

We now know that each point in the Fourier amplitude and phase spectrum corresponds to a complex exponential function with fixed amplitude and phase shift and oscillating with the omega frequency. Each point on the magnitude and phase surfaces in Laplace represents the magnitude and phase of a more general exponential function with exponent containing real and imaginary parts. Depending on whether it is positive or negative, the higher the sigma value, the greater the exponential growth or decline. The higher the omega, the higher the oscillation frequency. For real-valued time domain functions, these exponentials reduce to exponentially decaying sinusoids, since the imaginary parts cancel each other as we saw above.

So the Laplace transform is the general form of the Fourier transform. Decomposes any signal into a linear combination of exponential functions with a complex exponent of real and imaginary parts. For example, when sigma is equal to zero, the rectangular pulse decomposes into a linear combination from e to j omega t, where omega goes from -inf to +inf. And this graph represents the magnitude of each of those components. While at sigma equal to -1, the same rectangular pulse decomposes into a linear combination of e a 1+j omega t, the same applies to every sigma value within the ROC.

Now, why is it very important to decompose the signal into exponential functions like this? Why not decompose the signal into a linear combination of rectangular functions or triangular functions? What's so special about exponentials? Well, most of the systems we work with as engineers are linear and time-invariant or can be approximated as linear time-invariant systems. These systems are modeled using time-invariant linear differential equations. In the language of mathematicians, linear ordinary differential equations of the nth order with constant coefficients and zero initial conditions. Exponential functions are the only functions that do not change when subjected to differential and integral operators.

When integrated or differentiated, it returns the same function multiplied by some constant. When an operator operating on a function produces a constant multiplied by the function, the function is called the eigenfunction of that operator and the constant is called the eigenvalue. Then the exponential function is the proper function of the differential and integral operators. And, in general, the exponential function with complex exponent is a function of time-invariant linear differential equations. And the solution to such equations consists only of exponential functions. Note that the exponential function with complex exponent contains cosine and sine waveforms. The general solution theorem in linear DE tells us that if you have a differential equation of the nth order and it has n independent linear solutions, then these solutions are called the fundamental set analogous to the basis set in vector theory and any solution can be written as a linear combination of them.

A set of functions is said to be linearly independent on an interval if they are not scalar multiples of each other, for example, sin(t) and cos(t) are linearly independent because sin is not a number multiplied by the cosine. e a 3t and e a 5t are also linearly independent, while x and 5x are linearly dependent. Exponential functions are always linearly independent, so the general solution to any differential equation is a linear combination of exponentials and sinusoids. Just like in vector space in which any vector can be represented as a linear combination of the basis set x,y,z.

The fundamental set of the solution of the time-invariant linear differential equation is a set of exponential functions and any linear combination of these exponentials is a solution of the differential equation. Now back to Laplace. From the Laplace transform, any signal can be decomposed into a linear combination of exponential functions with complex exponents. For example, exp(-5t) can be decomposed into a linear combination of exp(0+jwt) which, by the way, is a Fourier transform. The same function can be decomposed into a linear combination of exp(-1+jwt) and can also be decomposed into a linear combination of exp(-2+jwt).

We can sweep sigma up and down to get all the allowed values ​​to get the entire Laplace spectrum. . But notice that when sigma is equal to -5 the signal exp(-5t) is decomposed into a single exponential function with an exponent of -5 +j0 and that is because the signal is now one of the elementary functions on this axis sigma = -5. At this point, the magnitude spectrum explodes to infinity. This infinite magnitude concentrated at a point on the S plane is called the pole; the magnitude response is polarized upward at this point. And this is what we expect because the Laplace transform of the exponential function is 1/(s-p) where p is the exponential coefficient.

It is a displaced rational function. Do you remember how we graph rational functions? To be exact, the absolute value of the rational function. Now, because S is composed of 2 variables, the Laplace transform of the exponential function is a rational function of 2 variables sigma and j omega. a rational surface. You can think of it as a rational function that revolves around its vertical asymptote. The value of the constant A controls the strength of the curvature. So each pole represents a single exponential function with exponent equal to the location of the pole in the S plane. And each pole has its own frequency response.

The cut in the omega j axis. The frequency responses of all poles are approximately the same with some small changes depending on the location and weight of each pole. For example, changing the pole location changes the frequency by 3 dB (the frequency at which the frequency response scales by one over root 2). This can be quite obvious by setting the frequency axis as a logarithmic scale. And yes, it is a low pass filter. The pole is a low-pass filter that passes low frequencies and attenuates higher frequencies. This is the main idea behind Bode plots. Now I'm not going to delve into Bode plots here in this video, but the bottom line is that the greater the distance between the pole and the origin, the greater the 3 dB frequency and therefore the greater the width of band.

After the 3 dB frequency, the frequency response decays approximately with a fixed slope of -20 dB per drop. Therefore, the response of any LTI system or the solution of the LTI differential equation is a linear combination of weighted exponentials, hence a linear combination of shifted rational functions in the s-domain. And the net frequency response is the sum of all the individual responses generated by each rational function. Now, the sum of those rational functions is also a rational function and the numerator function depends on the weights of the individual rational functions. The sum of rational functions of a variable is a rational function that consists of peaks and valleys.

The roots of the numerator of the resulting function are called zeros. The zeros are the places where the function goes to zero. And once again, the Laplace transform of the solution of the LTI differential equation is a rational function of 2 variables or a 3D surface consisting of poles and zeros. The poles come from the exponential functions themselves, while the location of the zeros depends on the location of the poles and the weights of the exponentials. The zeros are the effect of adding the individual rational functions of each exponential. The resulting function is called the transfer function of the LTI system described by LTI DE.

Each of these zeros is a shifted line with a certain slope. Its absolute value is like an inverted triangle, while in the S plane it is cone-shaped. The frequency response of zero is approximately the inverse of that of a pole. It can also be displayed on a frequency axis with logarithmic scale. So the poles and zeros have well-defined shapes and knowing their locations is enough to construct the entire Laplace spectrum. The poles and zeros are properties of the transfer function and, therefore, of the differential equation that describes the LTI system. So the Laplace transform is a powerful tool to solve LTI DE and that because differentiation in the time domain is multiplying by S in the S domain and integration in the time domain is analogous to dividing by S in the S domain Now, with non-zero initial conditions, the differential equation becomes an algebraic equation of the variable S.

And the solution is in the form of a polynomial over a polynomial in the domain S _ The transfer function _ From the fundamental theorem of algebra we know that each of these polynomials in the numerator and denominator can be factored into products of first-order terms so that the roots are real or complex conjugate pairs. The zeros are the roots of the numerator, while the poles are the roots of the denominator. If the degree in the polynomial of the numerator is less than that of the denominator, using partial fractions the solution can be decomposed into a linear combination of first-order rational functions with a constant numerator.

And the corresponding time domain is a linear combination of exponentials. We don't need to plot the entire Laplace spectrum. We only need to know the location of the poles and zeros and represent them with crosses and circles respectively in the s-plane. The poles and zeros, as well as the gain constant k, are sufficient to fully characterize the differential equation and provide a complete description of the system. . Let me explain you. Each point on the complex S plane behaves exactly like a two-dimensional vector. And the difference between any 2 points is also a two-dimensional vector. If we have a transfer function like this where zi and pi represent the location of each zero and each pole in the s plane respectively.

If we takeany point in the complex S plane, then each term in the numerator (s-zi) represents a vector from that zero to that point with magnitude Ni and phase theta I. Similarly, each term in the denominator (s-pi ) represent a vector from the pole to the point with magnitude of Di and phase of phi i. The magnitude and phase response of the transfer function at any point in the S plane are given by this formula. By sweeping S across the entire S plane we obtain the entire Laplace spectrum. Similarly, by sweeping S along the jw axis we obtain the Fourier spectrum.

Now To end this discussion, let's talk about stability. For a system to be stable, it must have a limited output for a limited input signal. Poles that are in the right half of the s plane. They correspond to exponentially increasing exponential functions with positive sigma values. Then, the poles of the right half-plane correspond to time domain signals that increase without limit. For a linear system to be stable, all its poles must lie within the left half of the s-plane. The system is called marginally stable if it has one or more poles located on the imaginary axis that correspond to sinusoidal components in the time domain that do not increase or decay.