Binary Sneeze

Posts Tagged ‘theorem’

Introduction:

ESD shows energy distribution in frequency spectrum. Its units are jouls/hz. Total Energy of signal is area under ESD of signal. This relation is provided by Parseval’s Theorem, which relates energy in time domain to the energy in frequency domain.

That is:

Where |X(f)|² is Read the rest of this entry »

Undersampling is generally known to have no benifits and is known to make the signals distorted. The main objective of this tutorial is to understand the phenomenon of sampling and the use of under sampling to extract a band-pass signal.

Section 1: Introduction

The sampling theorem states that for a limited bandwidth (band-limited) signal with maximum frequency fmax, the equally spaced sampling frequency fs must be greater than twice of the maximum frequency fmax, i.e.,

fs > 2•fmax

In order to have the signal be uniquely reconstructed without aliasing.

The frequency 2•fmax is called the Nyquist sampling rate. Half of this value, fmax, is sometimes called the Nyquist frequency.

When this theorem is not followed and fs < 2•fmax, the concept of undersampling arises.

This undersampling results aliasing of the signal, which causes many problems while the reconstruction of the signal. But a band-pass signal can be successfully retrieved back by using undersampling.

Section 2: Low Pass Sampling

Sampling (C to D)

Sampling is a principle that engineers follow in the digitization of analog signals. For analog-to-digital or discreet-to-continuous conversion, to result in a faithful reproduction of the signal slices called samples, of the analog waveform. The number of samples per second is called the sampling rate or sampling frequency fs.

A continuous time signal I(x) (where x is the time) can be sampled at a sampling rate of fs by replacing x with n/fs which implies that I[n] = I(n/fs)

Nyquist’s Theorem

The Nyquist’s Theorem is also known as the sampling theorem. Read the rest of this entry »