first order Archives

First order digital PLL for tracking constant phase offset

Considering a typical scenario where there might exist a small phase offset between local oscillator between the transmitter and receiver.

Figure 1 : Transmitter receiver with constant phase offset

In such cases, it might be desirable to estimate and track the phase offset such that the performance of the receiver does not degrade.

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Sigma delta modulation

In an earlier post, it was mentioned that delta modulator without the quantizer is identical to convolving an input sequence with $[1\ -1]$ . Let us first try to validate that thought using a small MATLAB example and using the delta modulator circuit shown in Figure 9.13a of DSP-Proakis [1].

% delta modulation
xn = sin(2*pi*1/64*[0:63]);
xhatn = 0; 

for ii = 1:length(xn)

dn = xn(ii) - xhatn;
dqn = dn; % no quantizing done, when quantizing is done: dqn = 2*(dn>0) - 1;
xqn = dqn + xhatn; 

% dump of transmitter variables
dnDump(ii) = dn;
dqnDump(ii) = dqn;
xqnDump(ii) = xqn;
xhatnDump(ii) = xhatn; 

xhatn = xqn; %  variable storing one-sample delayed version of xn

% receiver
rn = rn + dqn;
rnDump(ii) = rn;

end 


% implementation by convolving with [1 -1] 

d1n = conv(xn,[1 -1]);
diff = (dnDump - d1n(1:64))

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Zero-order hold and first-order hold based interpolation

In problem 9.14 of DSP-Proakis, the objective is to analyze the effect of zero-order interpolation and first-order interpolation to double the number of samples in the sinusoidal $x(n) = \cos(\frac{2\pi n}{N})\ \ \ 0 \le n \le N-1$

while keeping the sampling frequency $F_s = 1/T$ unchanged.

My take:

The first part of the problem (a) is to generate the sequence $y(n)$ having half the frequency of $x(n)$ . For zero-order interpolation, the interpolated samples can be generated by holding the current sample till the new sampling instant (Ref: Section 9.3.1 -Sample and Hold [1]). The impulse response of such a system is a rectangular function.

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