Additional Notes 3: Carrier Phase Estimation Notes

Objective: To generate a phase reference for synchronous demodulation of DSB-SC AM signals.

Received Signal: The noise-corrupted received signal is represented as:

r(t) = u(t) + n(t) = A m(t) \cos(2\pi fc t + \phic) + n(t)

Key Observations:

• The received signal $r(t)$ has a zero mean because the message signal $m(t)$ contains no DC component.

• The average power output from a narrowband filter tuned to frequency $f_r$ is zero, preventing direct extraction of the carrier signal from $r(t)$.

Signal Manipulation:

By squaring $r(t)$:
r^2(t) = A^2 m^2(t) \cos^2(2\pi fc t + \phic) + \text{noise}

Expands to:
= \frac{1}{2} A^2 m^2(t) + \frac{1}{2} A^2 m^2(t) \cos(4\pi fc t + 2\phic) + \text{noise}

Squaring introduces a spectral component at twice the carrier frequency, $2f_c$.

Bandpass Filtering:

To isolate the desired $2fc$ component, the squared signal goes through a narrowband filter tuned to $2fc$:

The mean output is a sinusoid at frequency $2fc$, with phase and amplitude given by: \text{Amplitude} = \frac{A^2 m^2(t) H(2fc)}{2}

Phase-Locked Loop (PLL) Components:

Structure: A PLL comprises:

• A multiplier

• A loop filter

• A voltage-controlled oscillator (VCO).

Input to the PLL is a sinusoid:
\text{Input} = \cos(4\pi fc t + \theta0)

Output from the VCO:
\text{Output} = \sin(4\pi f_c t + \theta),
where $\theta$ is the phase estimate.

Multiplication Result:

The product of input and output yields:
e(t) = \cos(4\pi fc t + \theta0) \sin(4\pi f_c t + \theta)

This produces both a low-frequency term and a term at four times the carrier frequency.

Loop Filter:

A low-pass filter extracts the low-frequency component,
G(s) = \frac{K}{1 + 5s}
where $K$ and time constants are design parameters.

VCO Output Relation:

The VCO output ($v(t)$) relates to the oscillator's instantaneous phase and is governed by:
\text{Instantaneous Phase} = 4\pi fc t + Kv v(t)

Closed-Loop System Under Noise:

The system’s performance becomes complex due to the nonlinearity introduced by the sine function of the phase difference.

PLL operation is linearized under small phase errors,
\sin(2(\theta - \phi)) \approx 2(\theta - \phi)

Transfer Function:

The closed-loop transfer function of the PLL reads:
H(s) = \frac{K G(s)/s}{1 + KG(s)/s}

With substitution:
H(s) = \frac{K}{s^2 + 25 \omegan s + \omega0^2}

Noise Effects on Phase Estimation:

Evaluating noise effects when the input signal is:
s(t) = A \cos(2\pi f_c t + \phi(t))

with additive noise leading to a phase error $\Delta \phi$.

Variance of Phase Error:

In noise-dominant conditions, the variance of the phase error is:
\sigma^2 \approx \frac{A^2 N0 B{neq}}{2}
The performance curves show that an increase in noise results in higher phase estimate variance.

Squaring Loop for Carrier Recovery:

The squaring operation produces a double frequency component but also amplifies noise:
y(t) = u^2(t) + 2u(t)n(t) + n^2(t)

Squaring Loss:

Squaring leads to an increase of phase error variance (squaring loss):
SL = \frac{1}{1 + \frac{B{bp}}{2B{neq}}}

Costas Loop:

A second method for generating a phase carrier to demodulate a DSB-SC signal is through a Costas loop, which uses phase-shifted multipliers.

Conclusion:

Understanding PLL mechanisms allows for improved phase synchronization in communication systems, crucial for effective signal demodulation, particularly in noisy environments.