Signal Transmission and Filtering

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Roadmap1. Response of LTI Systems2. Signal Distortion in Transmission3. Transmission Loss and Decibels4. Filters and Filtering5. Quadrature Filters and Hilbert Transforms6. Correlation and Spectral Density

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RESPONSE OF LTI SYSTEMS

• Impulse Response and the Superposition Integral• Transfer Functions and Frequency Response• Block-Diagram Analysis

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Impulse Response and the Superposition Integral

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The output y(t) is then the forced response due entirely to x(t)

where F[x(t)] stands for the functional relationship between input and output

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What is LTI means ?

The linear property means that the system equation obeys the principle of superposition. Thus, if

where ak are constants, then

The time-invariance property means that the system’s characteristics remain fixed with time. Thus, a time-shifted input x(t – td) produces

so the output is time-shifted but otherwise unchanged.

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Direct analysis of a lumped-parameter system starting with the element equations leads to the input–output relation as a linear differential equation in the form

Unfortunately, this Eq. doesn’t provide us with a direct expression for y(t)

To obtain an explicit input–output equation, we must first define the system’simpulse response function

which equals the forced response when x(t) = δ(t)

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Any continuous input signal can be written as the convolution x(t) = x(t)*δ(t) , so

From the time-invariance property, F[δ(t - λ)] = h(t – λ) and hence

superposition integral

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Various techniques exist for determining h(t) from a differential equation or some other system model. However, you may be more comfortable taking x(t) = u(t) and calculating the system’s step response

from which

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EXAMPLE: Time Response of a First-Order System

This circuit is a first-order system governed by the differential equation

From either the differential equation or the circuit diagram, the step response isreadily found to be

Interpreted physically, the capacitor starts at zero initial voltage and charges toward y(∞) = 1with time constant RC when x(t) = u (t)

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The corresponding impulse response

The response to an arbitrary input x(t) can now be found by putting the impulse response equation in the superposition integral.

Rectangular pulse applied at t = 0, so x(t) = A for 0 < t < τ .

The convolution y(t) = h(t) * x(t) divides into three parts, with the result that

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Transfer Functions and Frequency Response

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Time-domain analysis becomes increasingly difficult for higher-order systems, we got a clearer view of system response by going to the frequency domain.

As a first step in this direction, we define the system transfer function to be the Fourier transform of the impulse response, namely,

This definition requires that H(f) exists, at least in a limiting sense. In the case of an unstable system, h(t) grows with time and H(f) does not exist.

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When h(t) is a real time function, H(f) has the hermitian symmetry

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The steady-state forced response is

Converting H(f0) to polar form then yields

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if

then

Since Ay/Ax = |H(f0)| at any frequency f0,

|H(f)| represents the system’s amplitude ratio as a function of frequency (sometimes called the amplitude response or gain)

arg H(f) represents the phase shift, since φy – φx = arg H(f)

Plots of |H(f)| and arg H(f) versus frequency give the system’s frequency response

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let x(t) be any signal with spectrum X(f)

we take the transform of y(t) = x(t) * h(t) to obtain

The output spectrum Y(f) equals the input spectrum X(f) multiplied by the transfer function H(f).

If x(t) is an energy signal, then y(t) will be an energy signal whose spectral density and total energy are given by

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Other ways of determining H(f)

calculate a system’s steady-state phasor response,

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EXAMPLE: Frequency Response of a First-Order System

ZR = R and ZC = 1/jωC

y(t)/x(t) = ZC/(ZC + ZR) when x(t) = ejωt

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We call this particular system a lowpass filter because it has almost no effect on the amplitude of low-frequency components, say |f| << B , while it drastically reduces the amplitude of high-frequency components, say |f| << B

The parameter B serves as a measure of the filter’s passband or bandwidth.

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|H(f)| ≈ 1, and arg H(f) ≈ 0

If W << B

Thus,

so we have undistorted transmission through the filter.

over the signal’s frequency range |f| < W

Y(f) = H(f)X(f) ≈ X(f) and y(t) ≈ x(t)

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If W ≈ B

Y(f) depends on both H(f) and X(f).

We can say that the output is distorted, since y(t) will differ significantly from x(t), but time-domain calculations would be required to find the actual waveform.

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If W >> B

The output signal now looks like the filter’s impulse response. Under this condition, we can reasonably model the input signal as an impulse.

The input spectrum has a nearly constantvalue X(0) for |f| < B

Y(f) ≈ X(0)H(f), y(t) ≈ X(0)h(t)

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Our previous time-domain analysis with a rectangular input pulse confirms these conclusions since the nominal spectral width of the pulse is W = 1/τ. The case W << B thus corresponds to 1/τ << 1/2πRC or τ/RC >> 1, and y(t) ≈ x(t).

Conversely, W >> B corresponds to τ/RC << 1 where y(t) looks more like x(t).

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Block-Diagram Analysis

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When the subsystems in question are described by individual transfer functions, it is possible and desirable to lump them together and speak of the overall system transfer function.

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EXAMPLE: Zero-Order Hold

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To confirm this result by another route,

let’s calculate the impulse response h(t) drawing upon the definition that y(t) = h(t) when x(t) = δ(t)

The input to the integrator then is x(t) - x(t - T) = δ(t) - δ(t - T), so

Which represents a rectangular pulse starting at t = 0. Rewriting the impulse response as h(t) = ∏ [(t – T/2)/T] helps verify the transform relation h(t) ↔ H(f).