0.9 Carrier recovery (Page 7/19)

Software receiver design Page 7 / 19

J_{S D} (θ) = \frac{1}{4} LPF {{(r_{p} (k T_{s}) - cos (4 π f_{0} k T_{s} + 2 θ))}^{2}} .

Substituting $r_{p} (k T_{s})$ from [link] and assuming $f_{c} = f_{0}$ , this can be rewritten

= \frac{1}{4} LPF {{(cos (4 π f_{0} k T_{s} + 2 Φ) - cos (4 π f_{0} k T_{s} + 2 θ))}^{2}} .

Expanding the square gives

\begin{matrix} = \frac{1}{4} LPF {{cos}^{2} & (4 π f_{0} k T_{s} + 2 Φ) - 2 cos (4 π f_{0} k T_{s} + 2 Φ) \cdot \\ cos & (4 π f_{0} k T_{s} + 2 θ) + {cos}^{2} (4 π f_{0} k T_{s} + 2 θ)} . \end{matrix}

Using the trigonometric formula [link] for the square of a cosine and the formula [link] for the cosine angle sum (i.e., expand $cos (x + y)$ with $x = 4 π f_{0} k T_{s}$ and $y = 2 Φ$ , and then again with $y = 2 θ$ ) yields

\begin{matrix} J_{S D} (θ) & = & \frac{1}{8} LPF {2 + cos (8 π f_{0} k T_{s} + 4 Φ) - 2 cos (2 Φ - 2 θ) \\ - & 2 cos (8 π f_{0} k T_{s} + 2 Φ + 2 θ) + cos (8 π f_{0} k T_{s} + 4 θ)} . \end{matrix}

By the linearity of the LPF,

\begin{matrix} = \frac{1}{4} & + & \frac{1}{8} LPF {cos (8 π f_{0} k T_{s} + 4 Φ)} - \frac{1}{4} LPF {cos (2 Φ - 2 θ)} \\ - & \frac{1}{4} LPF {cos (8 π f_{0} k T_{s} + 2 Φ + 2 θ)} \\ + & \frac{1}{8} LPF {cos (8 π f_{0} k T_{s} + 4 θ)} . \end{matrix}

Assuming that the cutoff frequency of the lowpass filter is less than $4 f_{0}$ , this simplifies to

J_{S D} (θ) \approx \frac{1}{4} (1 - cos (2 Φ - 2 θ)),

which is shown in the top plot of [link] for $Φ = - 0.8$ . The algorithm [link] is initialized with $θ [0]$ at some point on the surface of the undulating sinusoidal curve. At each iterationof the algorithm, it moves downhill. Eventually, it will reach one of the nearby minima, which occur at $θ = - 0.8 \pm n π$ for some $n$ . Thus, [link] provides evidence that the algorithm can successfully locate the unknown phase, assuming that the preprocessed signal $r_{p} (t)$ has the form of (10.3).

The error surface Equation 17 for the SD phase tracking algorithm is shown in the top plot. Analogous error surfaces for the phase locked loop Equation 22 and the Costas loop Equation 27 are shown in the middle and bottom plots. All have minima (or maxima) at the desired locations (in this case -0.8) plus nπ offsets. — The error surface [link] for the SD phase tracking algorithm is shown in the top plot. Analogous error surfaces forthe phase locked loop [link] and the Costas loop [link] are shown in the middle and bottom plots. All have minima (or maxima) at the desired locations (in thiscase $-$ 0.8) plus $n π$ offsets.

[link] shows the algorithm [link] with the averaging operation replaced by the more general LPF. In fact, this provides a concrete answer to [link] ; the averaging, the LPF, and the integral block allact as lowpass filters. All that was required of the filtering in order to arrive at [link] from [link] was that it remove frequencies above $4 f_{0}$ . This mild requirement is accomplished even bythe integrator alone.

A block diagram of the squared difference phase tracking algorithm Equation 11. The input r_p(kT_s) is a preprocessed version of the received signal as shown in Figure 10-3. The integrator block Σ has a lowpass character, and is equivalent to a sum and delay as shown in Figure 7-5. — A block diagram of the squared difference phase tracking algorithm [link] . The input $r_{p} (k T_{s})$ is a preprocessed version of the received signal as shown in [link] . The integrator block $Σ$ has a lowpass character, and is equivalent to a sum and delay as shown in [link] .

The code in pllsd.m is simplified in the sense that the received signal rp contains just the unmodulated carrier. Implement a more realistic scenario by combining pulrecsig.m to include a binary message sequence, pllpreprocess.m to create rp , and pllsd.m to recover the unknown phase offset of the carrier.

Using the default values in pulrecsig.m and pllpreprocess.m results in a $ψ$ of zero. [link] provided several situations in which $ψ \neq 0$ . Modify pllsd.m to allow for nonzero $ψ$ , and verify the code for the cases suggested in [link] .

Investigate how the SD algorithm performs when the received signal contains pulse shaped 4-PAM data.Can you choose parameters so that $θ \to Φ$ ?

Consider the sampled cosine wave

x (k T_{s}) = cos (2 π f_{0} k T_{s} + α)

where the frequency $f_{0}$ is known but the phase $α$ is not. Form

v (k T_{s}) = x (k T_{s}) cos (2 π f_{0} k T_{s} + β (k))

using the current (i.e. at the $k$ th sample instant) estimate $β (k)$ of $α$ and define the candidate cost function

J (β) = LPF {v^{2} (k T_{s})}

where the cutoff frequency of the ideal lowpass filter is $0.8 f_{0}$ .

Does minimizing or maximizing $J (β)$ result in $β = α$ ? Justify your answer.
Develop a small stepsize gradient descent algorithm for updating $β (k)$ to estimate $α$ . Be certain that all of the signals needed to implement thisalgorithm are measurable quantities. For example, $x$ is directly measurable, but $α$ is not.
Determine the range of the initial guesses for $β (k)$ in the algorithm of part (b) that will lead to the desired convergence to $α$ given a suitably small stepsize.

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Source: OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3

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