FM Demodulation using Phase Locked Loop PLL

1
FM Demodulation using Phase Locked Loop (PLL)
where

• The VCO generates a sinusoid of a fixed frequency
  fc in the absence of an input control voltage.
  When there is a signal at the input of the VCO
  the instantaneous frequency of the VCO is

2
FM Demodulation using PLL
where kv is the deviation constant. The VCO
output can be written as
where

• The phase compensator block is basically a
  multiplier and a filter that rejects signal
  components centered at 2fc. Hence its output may
  be expressed as

3
FM Demodulation using PLL
where the difference
constitutes the phase error. The signal
e(t) is the input to the loop filter.

• Lets assume that the PLL is in lock so that the
  phase error is small. Then, we have

under this condition we may deal with the
linearized model of PLL
4
FM Demodulation using PLL
or equivalently as
or
Taking FT, we have
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FM Demodulation using PLL
Suppose that we design G(f) such that
in the frequency band of the message
signal. Then, we have
6
Sampling

• Theorem Let the signal x(t) be bandlimited with
  bandwidth W i.e., let X(f)0 for . Let
  x(t) be sampled at multiples of some basic
  sampling intervals Ts, where to
  yield
• the sequence . Then it
  is possible to reconstruct the original signal
  x(t) from the sampled values by the
  reconstruction formula

where is any arbitrary number that satisfies
in the special case where we have
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Sampling

• PROOF Let denote the result of sampling
  the original signal by impulses at nTs time
  instants. Then, we have

Taking FT, we have
8
Sampling

• If , then the replicated spectrum
  of x(t) overlaps and the reconstruction of the
  original signal is not possible. This is called
  aliasing.

• To obtain the original signal one obvious choice
  is to pass the sampled signal through a LFP.

9
Sampling
with this choice, we have
taking IFT, we have
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Practical Sampling

• Switched/Natural Sampling In switched sampling,
  the bandlimited signal is sampled by closing a
  switch at regular intervals and keeping it close
  for some fixed duration.

and the sampled signal xs(t) can be written as
taking FT, we have
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Natural Sampling
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Zero-Order-Hold Sampling

• Zero-Order-Hold Flat top or zero-order-hold
  (ZOH) sampling is done by sampling at nTs and
  holding the result until the time of the next
  sample i.e., (n1)Ts. Zero-order-hold sampling
  can be considered as a cascade of impulse
  sampling with a zero-order-hold filter i.e. a
  filter with a impulse response of

taking FT, we have
output of the impulse sampler in frequency domain
is
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Zero-Order-Hold Sampling
passing this through ZOH filter we have
to obtain the original signal back, a filter with
the following frequency response should be used

• If sampling is quite high, that is, fs gtgt 2W,
  then for all fltW, Tsf ltlt1 and sinc(Tsf) is
  approximately 1. In this case an ideal filter can
  reconstruct the original signal with good
  accuracy.