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# LTI Systems Characterized by LCCDEs

If an LTI system can be described by an LCCDE in time domain

then after taking Laplace transform of the LCCDE, it can be represented as an algebraic equation in the domain

and its transfer function is rational

where is a constant and are the zeros of (roots of the numerator polynomial ) and are the poles of (roots of the denominator polynomial ). The LCCDE alone does not completely specify the relationship between and , as additional information such as the initial conditions is needed. Similarly, the transfer function does not completely specify the system. For example, the same with different ROCs will represent different systems (e.g., causal or anti-causal).

Example 1: A circuit consisting an inductor and a resistor with input voltage applied to the two element in series can be described by an LCCDE:

Taking Laplace transform of this equation, we get

• If the output is the current through the RL circuit, then the ratio between the input and output is defined as the conductance of the circuit:

where and is the impedance of the circuit. In time domain, we have:

• If the output is the voltage across the resistor , then the transfer function of the system (a voltage divider) is

where . In time domain, the impulse response of the system is

• If the output is the voltage across the inductor , then the transfer function is

with impulse response in time domain:

As the ROCs of both and are the same half plane to the right of the only pole on the negative side of the real axis, the -axis is contained in ROC and the corresponding frequency response function exists:

Example 2: A voltage is applied as the input to a resistor , a capacitor and an inductor connected in series. According to Kirchhoff's voltage law, the, the system can be described by a differential equation in time domain:

or an algebraic equation in s-domain:

If the current through the circuit is treated as the output, then the transfer function of the system is

which is the overall impedance of the circuit composed of the individual impedance of the three elements

 resistor capacitor inductor time domain s-domain impedance
If the output is the voltage across one of the three elements (, , or ), the transfer function can be easily obtained by treating the series circuit as a voltage divider:
• Output is voltage across the capacitor

• Output is voltage across the resistor

• Output is voltage across the inductor

If we define

the common denominator of the transfer functions can be written in standard (canonical) form

with two roots

and the transfer functions above can be written in standard forms:
• Output across C:

with two poles and no zeros.
• Output across R:

with two poles and one zero at the origin.
• Output across L:

with two poles and two repeated zeros at the origin.
As to be discussed later, the magnitude and phase of the corresponding frequency response function can be qualitatively determined in the s-plane, and it turns out that the three transfer functions behave like low-pass, band-pass and high-pass filter, respectively. Moreover, when the common real part of the two complex conjugate poles is small (i.e., ), there will be a narrow pass-band around in all three cases.

Example 3: System identification: find and of an LTI, based on the given input and output :

In s-domain, input and output signals are

The transfer function can therefore be obtained

This system has two poles and and therefore there are three possible ROCs:
• , is left sided (anti-causal, unstable);
• , is two sided (non-causal, unstable);
• , is right sided (causal, stable).
We need to determine which of these ROCs is true for . As the ROC of a product is the intersection of the ROCs of the factors (without zero-pole cancellation):

ROC of must be the third one above, and we have:

The equation for above can be written as:

Its inverse Laplace transform is the LCCDE of the system:

Next: Evaluation of Fourier Transform Up: Laplace_Transform Previous: Representation of LTI Systems
Ruye Wang 2012-01-28