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.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

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progressive wave

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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

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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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