Noise in Continuous Wave CW Modulation

1
Noise in Continuous Wave (CW) Modulation

• To study the effects of channel noise on the
  reception of CW modulated signals we will
  formulate two necessary models
• Channel Model This model assumes that a
  communication channel that is distortionless but
  perturbed by additive white Gaussian noise
  (AWGN).
• Receiver Model The receiver model assumes that
  the receiver consists of an ideal bandpass filter
  (BPF) followed by an ideal demodulator
  appropriate for the application at hand. The BPF
  is used to minimize the effect of channel noise.

2
Noise in Continuous Wave (CW) Modulation

• Signal-to-noise ratios Let the power spectral
  density (PSD) of the noise w(t) be denoted by
  N0/2, which is defined for both positive and
  negative frequencies, that is N0 is the average
  noise power per unit bandwidth measured at the
  front end of the receiver. The BPF is assumed to
  be ideal having a bandwidth B and a midband
  frequency equal to the carrier frequency fc. The
  later assumption is justified for DSB-SC, AM and
  FM whereas, for SSB and VSB modulation special
  considerations are required.

3
Noise in Continuous Wave (CW) Modulation
where nI(t) is the in-phase component and nQ(t)
is the quadrature noise component. The filtered
signal x(t) available for demodulation is defined
by

• The details of s(t) depend on the type of
  modulation used. The noise power is N0B.

• The input signal-to-noise ratio (SNR)I is defined
  as the ratio of the average power of the
  modulated signal s(t) to the average power of the
  filtered noise n(t).

• The output signal-to-noise ratio (SNR)O is
  defined as the ratio of the average power of the
  demodulated message signal to the average power
  of the noise, both measured at the receiver
  output.

4
Noise in Continuous Wave (CW) Modulation

• In order to compare the performance of different
  modulation schemes it is essential that it must
  be done on an equal basis as described here
• The modulated signal s(t) transmitted by each
  system has the same average power.
• The channel noise w(t) has the same average power
  measured in the message bandwidth W.

• As a frame of reference the channel
  signal-to-noise ratio, (SNR)C is defined as the
  ratio of the average power of the modulated
  signal to the average power of the channel noise
  in the message bandwidth, both measure at the
  receiver input.

5
Noise in Continuous Wave (CW) Modulation

• For the purpose of comparing different CW
  modulation systems, we normalize the receiver
  performance by dividing (SNR)O by (SNR)C. This
  ratio is called figure of merit for the receiver
  and is defined as

• The higher the value of the figure of merit the
  better the noise performance of the receiver will
  be.

6
Noise in Linear Receivers using Coherent Detection

• Lets consider the case of DSB-SC. The expression
  for the modulated signal is given as

• We assume the message m(t) to be a sample
  function of a stationary process of zero mean
  with PSD SM(f) having a bandwidth of W. The
  average power of the message is

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Noise in Linear Receivers using Coherent Detection

• The carrier wave is statistically independent of
  the message signal. The average power of DSB-SC
  modulated
• component of s(t) is , with a
  noise PSD of N0/2 the
• average noise power in the message
  bandwidth W equals WN0 (baseband scenario).
  Hence, we have

where Pm is the power of the message
Finding an expression for (SNR)O, we have
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Noise in Linear Receivers using Coherent Detection
Output of LPF

• The power of the signal component at the receiver
  output is . The average power of the
  filtered noise is 2WN0.

Remembering that

• The average noise power at the receiver output is

Hence, we have
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Noise in AM Receivers using Envelope Detection

• The expression for AM signal is given as

where it is assumed that
10
Noise in AM Receivers using Envelope Detection

• The average power of the carrier in the AM signal
  s(t) is The average power of the information
  bearing component
• is
  . Average power of the full AM signal
  s(t) is . The average
  power of the noise in the message bandwidth is
  . Hence, the channel signal to noise ratio
  for AM is

Finding an expression for (SNR)O, we have
using phasor representation
11
Noise in AM Receivers using Envelope Detection

• The phase of x(t) is of no interest as an ideal
  envelope detector is insensitive to the phase
  variations in x(t).
• The analysis of the above expressions is complex
  and needs to be simplified to obtain a simplified
  result in the following format

• When the average carrier power is large as
  compared to the average noise power, the receiver
  is operating satisfactorily, then the signal term
  will be large as compared to
  and . Then y(t) can be approximated
  as

usually the DC term is not part of the message
and can be blocked by passing the signal through
a capacitor. So, the DC term can be neglected.
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Noise in AM Receivers using Envelope Detection
Hence, we have
13
Threshold Effect

• When carrier-to-noise ratio is small as compared
  to unity the noise term dominates the performance
  of the envelope detector and is completely
  different. Representing the narrowband noise n(t)
  in terms of its envelope and phase, we have

• The phasor diagram for x(t) s(t) n(t) becomes

14
Threshold Effect

• The noise envelope is used as a reference here
  due to its dominance. Here it is assumed that Ac
  is small as compared to r(t). If we neglect the
  quadrature component of the signal with respect
  to the noise we have

• Hence, when carrier-to-noise ratio is small the
  detector has no component that is strictly
  proportional to the message signal m(t).
  Recalling that is uniformly distributed
  over radians. Hence, it follows that we have a
  complete loss of information at the detector
  output (as expected value will be zero). This
  loss of information m(t) at the output of the
  envelope detector is called the threshold effect.

• Threshold effect means a value of the
  carrier-to-noise below which the noise
  performance deteriorates much faster than
  proportionately to the carrier-to-noise ratio.

15
Threshold Effect

• Every non-linear detector has a threshold effect
  whereas, coherent detectors do not suffer from
  threshold effect.

• Carrier-to-ratio In order to find this quantity
  we make the information part to be zero. Hence,
  we have

where is the unmodulated carrier
and n(t) is the sample function of band-limited,
zero mean, white Gaussian noise.

• The input signal to noise ratio consists of an
  unmodulated caarier with an average power equal
  to and the average noise power at the
  detector input is

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Carrier-to-Noise Ratio

• The carrier-to-noise ratio is therefore

where can be thought of as an input
signal-to-noise ratio.

• Envelope detector favors strong signals and
  penalize weak signals. This phenomenon is
  referred to as weak signal suppression which is a
  manifestation of the threshold effect.