# Response of a Ltic System Time-Domain Analysis

Topics: LTI system theory, Digital signal processing, Impulse response Pages: 9 (1645 words) Published: June 24, 2013
Circuits & Signals
EEE/ INSTR C272
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ANU GUPTA EEE

Time-domain analysis
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Response of a LTIC system
time-domain analysis
linear, time-invariant, continuous-time (LTIC) systems--Total response = zero-input response + zero-state response zero-input response component that results only from the initial i t t th t lt l f th i iti l conditions at t = 0 with the input f(t) = 0 for t ≥ 0, zero-state zero state response component that results only from the input f(t) or t ≥ 0 when the initial conditions (at t = 0) are assumed to be zero. decomposition property. The system output is zero when the input is zero only if the system is in zero state. Zero state, meaning the absence of all internal energy g ; , storages; that is, all initial conditions are zero.

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LTIC System C. C T

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Differentiator increases noise Why?
The derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity Derivative of a fluctuating signal has more fluctuations then the signal itself . As it shows change in a small interval Δt. A steep change causes derivative to show large peaks graphically

Thus if noise ( random signal) in a signal is getting differentiated along with the signal then resultant signal has more noise signal,

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Differentiation of a signal has the reputation for being a noisy operation. Even if the signal is band-limited, noise will introduce high frequency signals which are greatly amplified by differentiation. p y Thus in a differentiator , high-frequency noise signals will not be suppressed by this circuit; rather they will be amplified f b lifi d far beyond th amplification of th d i d signal d the lifi ti f the desired i l Thus to smoothen out the signal filters are used which signal, integrates the signal (i.e. takes average of the signal)

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PROOF

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R c differentiatorfreq. selective operation q p

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Zero input response
input f(t) = 0 so that

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Physics of natural modes
The system uses a proper combination of characteristic modes to come back to the rest position while satisfying appropriate boundary (or initial) conditions. Any combination of characteristic modes can be sustained by the system alone without requiring an external input.

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Example1
Distinct Roots

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Example2
Complex Roots

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Zero state response
To determine the response of a linear system to an arbitrary input f(t) f(t). The impulse function δ(t) is used in determining the response of a linear system to an arbitrary input f(t). y y p () Approximating f(t) with narrow rectangular pulses. The system response to the input f(t) is then given by the sum of the system's responses to each (delayed) impulse system s component of f(t).

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LTI system response to any signal

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input f(t) represented as a sum  of all the impulse components

output y(t) represented as a sum  of the output components

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Graphical Representation impulse response

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Zero state response
For linear, time‐invariant (LTI) system

We have obtained the system response yet) to input y p y ) p f(t) in terms of the unit impulse response h(t). Knowing h(t), we can determine the response y (t) to any...