We will start with the same impulse response of a linear, time-invariant
system used in the Fourier Transform example, which is given by
the following equation and shown in the accompanying figure.

Impulse Response h(t)

Since h(t) is 0 for all negative time and is a sum of exponentials and time-weighted exponentials, the one-sided Laplace Transform can be taken for each term. The overall Laplace Transform is the sum of the individual terms, making use of the linearity property. Using the defining integral for the Laplace Transform or using the Tables of Laplace Transforms from the textbook, the individual transforms and the total transform for this system are shown below. The region of convergence for the Laplace Transform is all of the s-plane to the right of Re[s] = -0.1.

H(s) is known as the **transfer function** for the system. Since the
region of convergence for H(s) includes the jw axis, we can
evaluate H(s) along the jw axis and get the Fourier Transform from the
previous example. That expression is repeated below; the magnitude and
phase plots are in the figure below. Compare H(w) and H(s). Note that the
magnitude and phase curves are on the same plot. The numbers on the
vertical axis represent decibels for the magnitude curve, and they
represent degrees for the phase curve.

Magnitude and Phase Plots

The MATLAB function "residue" can be used to do partial fraction
expansion. Given the numerator and denominator polynomials for a
transfer function H(s), the syntax is

where P is a vector containing the poles of the transfer function, R is a vector containing the coeficients (residues) associated with the poles, and K is the direct feedthrough term from input to output. If the degree of the denominator is greater than that of the numerator (strictly proper transfer function), then K is 0. Each coefficient in R is associated with the pole in the corresponding element of P. For repeated roots, the first element is that root raised to the 1st power, the second element is that root raised to the 2nd power, etc. In this example, the vector P will be [-5 -5 -2 -0.1], and the R vector will be [0 6 5 3]. K will be 0.

The roots of the numerator polynomial of the transfer function are the
**zeros** of the transfer function. The roots of the denominator
polynomial are the **poles** of the transfer function. The next figure
shows a plot of the poles (shown by x's) and zeros (shown by o's) for
H(s). The "(2)" below the pole at s = -5 indicates that there are two poles
at the same location. The transfer function can be evaluated at any point
in the s-plane other than at its poles (singularities). The plot also
shows a point s = s_{1} = -2+j3.

Poles and Zeros of H(s)

The magnitude and phase of H(s) at a point s=s1 can be obtained by
substituting s = s_{1} into H(s) and computing the magnitudes and phases of the
various complex numbers which are obtained. The overall magnitude is the
magnitude of the numerator divided by the magnitude of the denominator, and
the overall phase is the phase of the numerator minus the phase of the
denominator. Graphically, the magnitude of any pole or zero evaluated at s = s_{1} is the length of a vector drawn from the pole or zero to the point
s_{1}.
The phase of the pole or zero at s_{1} is the angle that the vector from the
pole or zero to s_{1} makes with respect to the positive real axis, with
positive angles measured counter-clockwise from the positive real axis.
The vectors are shown from the poles and zeros of H(s) to the point s_{1} in
the next figure. The magnitude and phase values shown in the figure are
the magnitude and phase of H(s) evaluated at s = s_{1}.

Vectors to s_{1}

The frequency response of the transfer function is obtained by evaluating
H(s) for values of s on the jw axis, that is, s = jw. This is shown
graphically in the next figure for s = s_{2} = j3.02 (w = 3.02 is the closest
value to 3.0 appearing in the frequency vector). The magnitude and phase
values shown in the figure are those which would be obtained from the
"bode" function in MATLAB or by substituting that value of frequency into
the Fourier Transform for this system. This is illustrated in the last
figure which repeats the magnitude and phase curves from the first figure
and superimposes the magnitude and phase at the point s = s_{2 }= j3.02.

Vectors to s_{2}

Magnitude and Phase repeated with s_{2}

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