Signals & Systems, Laplace Transform, 3

The transfer function H(s) is defined under what initial conditions?

All initial conditions are assumed to be zero. Nonzero conditions are handled using the zero- input response.

How do we convert a function of form H(ω) into one of form H(s)?

Assuming H(s) is a rational function (ratio of two polynomials), H(s) and H(ω) are related by s = jω. Replace -ω2 with s2 and jω with s everywhere. Equivalently, replace ω = -j(jω).

Explain the similarities and differences between the time-shift and frequency-shift properties of the Laplace transform.

A shift by T in the time domain becomes a multiplication by e-Ts in the frequency domain, and a shift by a in the frequency domain becomes a multiplication by e-at in the frequency domain.

Is the uniqueness property of the Laplace transform unidirectional or bidirectional? Why is that significant?

Bidirectional, provided we restrict the time domain signal x(t) to be causal. It can be shown that the bilateral Laplace transforms of e-at u(t) and -eat u(-t) are both 1/(s + a), so only if x(t) = 0 for t < 0 is uniqueness guaranteed. The significance is that inverse Laplace transforms can be computed using a table lookup, instead of the complex integral given by Eq. (3.5).

When evaluating the expansion coefficients of a function containing repeated poles, is it more practical to start by evaluating the coefficient of the fraction with the lowest-order pole or that with the highest-order pole? Why?

Highest order. See the procedure below:

Why doesn't a strictly proper transfer function have a BIBO stable and causal inverse system?

If H(s) is strictly proper, then the numerator polynomial of H(s) has smaller degree than the degree of the denominator polynomial. Then the transfer function of the inverse system G(s) = 1/H(s) is strictly improper, and must be unstable (see Concept Question 3-10).

For stable systems, the forced response includes what other responses?

It includes the transient response and the steady-state response. See Eq. (3.162) for an example. The first term is the steady-state response and the second term is the transient response.

Is convergence of the Laplace transform integral an issue when applied to physically realizable systems? If not, why not?

No, because the Laplace transform integral of x(t) converges provided |x(t)| does not increase faster than an exponential function. Physically realizable systems do not create outputs that increase faster than exponential functions. See Eq. (3.4).

Does knowledge of just the poles and zeros completely determine the LCCDE?

No. The constant A above also is needed.

According to the time scaling property of the Laplace transform, "shrinking the time axis corresponds to stretching the s-domain." What does that mean?

The Laplace transform of x(at) is X(s/a). Multiplication of t by a shrinks x(t), Division of s by a expands X(s).

Why is a system with an improper transfer function always BIBO unstable?

The partial fraction expansion of an improper H(s) includes a term proportional to s. See Eq. (3.104) for an example. Consider a bounded input x(t) = u(t). Then X(s) = 1/s and Y(s) = H(s) X(s) includes a constant term, whose inverse Laplace transform is an impulse, so the output is unbounded.

Why is it that zeros of the transfer function have no bearing on system stability?

The partial fraction expansion of the transfer function is a sum of terms of forms Ci /(s - pi) where pi are its poles. Its zeros affect only the constants Ci. The inverse Laplace transform of the transfer function, the impulse response, is a sum of terms Ci epit u(t), and the system is stable only if all of the pi are in the left half-plane.

What purpose does the partial fraction expansion method serve?

The partial fraction expansion of the transfer function is a sum of terms of the forms Ci /(s - pi) where pi are its poles and Ci are its residues. The inverse Laplace transform of the transfer function, which is the impulse response, is then a sum of terms Ciepit u(t).

How does one determine the poles and zeros of a rational function X(s)?

The poles of X(s) are the roots of the denominator polynomial set equal to zero. The zeros of X(s) are the roots of the numerator polynomial set equal to zero.

For stable systems, the transient response includes what other responses?

The transient response includes the natural response. It may also include part of the forced response, if the input is a decaying exponential function. See Eq. (3.144) for an example of this:

What role do zeros of transfer functions play in system invertibility?

The zeros of transfer function H(s) are the poles of the transfer function of the inverse system G(s) = 1/H(s). So a stable inverse system exists only if the zeros of H(s) are all in the left half- plane.