March 2007 - DSP LOG

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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Signal to quantization noise in quantized sinusoidal

In problem 4.37 of DSP-Proakis [1], the task is to analyze the total harmonic distortion in quantized sinusoidal, $x(n) = \sin(2\pi f_0 n)$ where $f_0 = k/N$ .

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Harmonic distortion in digital sinusoidal generators

In Problem 4.36 of DSP-Proakis [1], the task is to provide insights into harmonic distortion which may be present in practical sinusoidal generators. Consider the signal
$x(n) = \cos(2\pi f_0 n)$ , where $f_0 = k_0/N$ .

My take:
The discrete time signal of fundamental period $N$ can consist of frequency components separated by $2\pi/N$ radians or $f =1/N$ cycles (Refer Section4.2 in [1]).

The Fourier series coefficient at frequency $f_0$ is,
$c_{k0} = \frac{1}{N}\sum_{n=0}^{N-1}x(n)e^{\frac{-j2\pi k_0 n}{N}}$ .

The total power of the signal over a period $P_x$ is same as the sum of the power of all the Fourier series coefficients (harmonic components) i.e.
$P_x = \frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2 = \sum_{k=0}^{N-1}|c_k|^2$ .

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LaTex on blogger not working?

I noticed that the scripts provided from http://wolverinex02.googlepages.com/emoticonsforblogger2 for writing $\LaTeX$ equations on blogger stopped working.