1. Signal Transmission
and Filtering

2. Roadmap
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1. Response of LTI Systems
2. Signal Distortion in Transmission
3. Transmission Loss and Decibels
4. Filters and Filtering
5. Quadrature Filters and Hilbert Transforms
6. 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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4. 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 4

5. What is LTI means ?
The linear property means that the system equation obeys the principle of
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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
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so the output is time-shifted but otherwise unchanged.

6. 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
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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’s
impulse response function
which equals the forced response when x(t) = δ(t)
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7. Any continuous input signal can be written as the convolution x(t) = x(t)*δ(t) ,
so
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From the time-invariance property, F[δ(t - λ)] = h(t – λ) and hence
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superposition integral

8. 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
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system’s step response
from which
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9. EXAMPLE: Time Response of a First-Order System
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This circuit is a first-order system governed by
the differential equation
From either the differential equation or the circuit diagram, the step response is
readily found to be
Interpreted physically, the capacitor starts at
zero initial voltage and charges toward y(∞) = 1 9
with time constant RC when x(t) = u (t)

10. The corresponding impulse response
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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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12. 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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13. When h(t) is a real time function, H(f) has the hermitian symmetry
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14. The steady-state forced response is
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Converting H(f0) to polar form then yields
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15. if
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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)
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Plots of |H(f)| and arg H(f) versus frequency give the system’s frequency response

16. let x(t) be any signal with spectrum X(f)
we take the transform of y(t) = x(t) * h(t) to obtain
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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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18. EXAMPLE: Frequency Response of a First-Order System
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y(t)/x(t) = ZC/(ZC + ZR) when x(t) = ejωt
ZR = R and ZC = 1/jωC
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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
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The parameter B serves as a measure of the filter’s passband or bandwidth.

20. If W << B
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|H(f)| ≈ 1, and arg H(f) ≈ 0
over the signal’s frequency range |f| < W
Thus,
Y(f) = H(f)X(f) ≈ X(f) and y(t) ≈ x(t)
so we have undistorted transmission through the filter.
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21. If W ≈ B
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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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22. If W >> B
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The input spectrum has a nearly constant
value X(0) for |f| < B
Y(f) ≈ X(0)H(f), y(t) ≈ X(0)h(t)
The output signal now looks like the filter’s
impulse response. Under this condition, we
can reasonably model the input signal as an
impulse.
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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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24. 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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26. EXAMPLE: Zero-Order Hold
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27. To confirm this result by another route,
let’s calculate the impulse response h(t) drawing upon the definition that
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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).
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