Linear Systems

1
Linear Systems

• Topics
• Review of Linear Systems
• Linear Time-Invariant Systems
• Impulse Response
• Transfer Functions
• Distortionless Transmission

2
Linear Time-Invariant Systems

• An electronic filter or system is Linear when
  Superposition holds, that is when,

• Where y(t) is the output and x(t)
  a1x1(t)a2x2(t) is the input.
• l. denotes the linear (differential equation)
  system operator acting on ..

• If the system is time invariant for any delayed
  input x(t t0), the output is delayed by just
  the same amount y(t t0).
• That is, the shape of the response is the same
  no matter when the input is applied to the system.

3
Impulse Response
x(t)

4
Impulse Response

• The linear time-invariant system without delay
  blocks is described by a linear ordinary
  differential equation with constant coefficients
  and may be characterized by its impulse response
  h(t).
• The impulse response is the solution to the
  differential equation when the forcing function
  is a Dirac delta function.
• y(t) h(t) when x(t)
  d(t).
• In physical networks, the impulse response has to
  be causal.
• h(t) 0 for t lt
  0
• Generally, an input waveform may be approximated
  by taking samples of the input at ?t-second
  intervals.
• Then using the time-invariant and superposition
  properties, we can obtain the approximate output
  as
• This expression becomes the exact result as ?t
  becomes zero. Letting n?t ?, we obtain

5
Transfer Function

• The output waveform for a time-invariant network
  can be obtained by convolving the input waveform
  with the impulse response of the system.
• The impulse response can be used to characterize
  the response of the system in the time domain.
• The spectrum of the output signal is obtained by
  taking the Fourier transform of both sides. Using
  the convolution theorem,

• Where H(f) Ih(t) is transfer function or
  frequency response of the network.
• The impulse response and frequency response are
  a Fourier transform pair
• Generally, transfer function H(f) is a complex
  quantity and can be written in polar form.

6
Transfer Function

• The H(f) is the Amplitude (or magnitude)
  Response.
• The Phase Response of the network is

• Since h(t) is a real function of time (for real
  networks), it follows
• H(f) is an even function of frequency and
• ?(f) is an odd function of frequency.

7
Power Transfer Function

• Derive the relationship between the power
  spectral density (PSD) at the input, Px(f), and
  that at the output, Py(f) , of a linear
  time-invariant network.

Using the definition of PSD
PSD of the output is
Using transfer function in a formal sense, we
obtain
Thus, the power transfer function of the network
is
8
Example RC Low Pass Filter
above fig.
9
Example 2.14 RC Low Pass Filter
the earlier found equations
later.
10
Example 2.14 RC Low Pass Filter
3dB Point
11
Distortionless Transmission
No Distortion if y(t)Ax(t-Td)
Y(f)AX(f)e-j2?fTd

• For no distortion at the output of an LTI system,
  two requirements must be satisfied
• The amplitude response is flat.
• H(f) Constant A ( No Amplitude
  Distortion)
• The phase response is a linear function of
  frequency.
• ?(f) ltH(f) -2pfTd (No Phase
  Distortion)

Second requirement is often specified
equivalently by using the time delay. We define
the time delay of the system as

• If Td(f) is not constant, there is phase
  distortion,
• Because the phase response ?(f), is not a linear
  function of frequency.

12
Example Distortion Caused By a filter

• For f lt0.5f0, the filter will provide almost
  distortionless transmission.
• The error in the magnitude response is less
  than 0.5 dB.
• The error in the phase is less than 2.18
  (8).
• For f ltf0,
• The error in the magnitude response is less
  than 3 dB.
• The error in the phase is less than 12.38
  (27).
• In engineering practice, this type of error is
  often considered to be tolerable.

13
Example Distortion Caused By a filter