The definition of Toeplitz matrix from [1] is:
A matrix is said to be Toeplitz if the elements
are determined completely by the difference
.
Month: April 2007
2nd order sigma delta modulator
In a previous post, the variance of the in-band quantization noise for a first order sigma delta modulator was derived. Taking it one step furhter, let us find the variance of the quantization noise filtered by a second order filter.
With a first order filter, the quantization noise passes through a system with transfer function (Refer Eq. 9.2.17 in [1]). The frequency response of
is
(Refer Eq. 9.2.18 in [1]).
Sigma delta modulation
In an earlier post, it was mentioned that delta modulator without the quantizer is identical to convolving an input sequence with . 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))