Analog-to-digital Converter

Analog-to-digital Converter (ADC) is the process of converting a continuous signal from the real-word to a digital signal  useful for a computer. A digital signal is sampled and quantized. The Digital-to-analog-Converter (DAC) is the conversion of a digital signal to a continuous signal. 

The ADC takes place in two steps: 

1. Sampling: convert the independent variable to discrete, for example consider the signal every 0.01 ms. This first stage is done by a Sample-and-Hold circuit. The sample rate must be choosen according to the sampling theorem, otherwise we will experiment aliasing (frequencies overlaps) producing changes in the signal's frequencies and even phase. The sampling theorem (Nyquist-Shannon theorem) stipulates that the sampling rate must be at least twice the highest frequency component of the analog signal. The half of the sampling frequeny is generally called Nyquist frequency or Nyquist rate.
2. Quantization: convert the measured or dependent variable to discrete variable.This second stage is processed by a n-bits ADC. For a 8 bit, we have 256 possible digital numbers to assign to a measured value.  Introducing a quantization error, generally uniformly distributed. The precision of the data decreasing with increasing bits number. In case of slowly varying signals, it is  showen that  adding  a  low random noise to the signal, we get more stable information about the underlying process. This operation is called Dithering. 

@uthor: Joan MOUBA. 

Statistics and Probability in DSP

In DSP, Probability is used to describe the process that generated the acquired signals, and Statistics are used to characterize the signals. The 2 approaches work by measuring some parameters from the acquired signals.  

The 2 most important statistics values are the mean and the standard deviation.

The mean is a measure of the average of all samples of the signal. The standard deviation is a measure of the  average of how the signal samples deviate from the mean. The square of the standard deviation is called variance. In denoising applications, the square of the mean could be addressed as the energy of the useful signal part and the variance as the noisy (undesired) signal part.

In DSP, it is important to make the difference between the characteristics of the acquired signals and the ones  of the underlying process. In other words, a underlying process may describe a model of a expected case and a acquired signal, a sample of that model. A process which characteristics vary other time is called a   nonstationary process, otherwise a stationary process.

Other process characteristics are:

• Histogram: Group the samples of a acquired signal that have the same value. it represents he number of occurences of a value in the signal 
  
• The probability mass function (pmf) is the ideal histogram representation computed by assuming a infinite number of sample, this is the histogram of the underlying process. The pmf is computable for discrete signals. The pmf has a discrete form.
  
• The probability distribution function or probability density function (pdf) is the pmf version for continuous signal. In difference the values of the pdf are normalized by the sample number, providing probability values lying between 0 and 1 and the area under the curve sum to one (1). The pdf has a continuous shape.
  
• Binning is used in case to fix a convenient number of bins (groups) in a histogram in order to have a reasonable representation.
  
• Cumulative distribution function (cmf) represents the probability that a value of the pdf lies between a certain range of values.  
• A very important distribution in DSP is the normal distribution, also called Gauss distribution or simply Gaussian. The Gaussian has a bell like shape with tails that fall quick to zero. The Gaussian function is fully charaterized by the mean and the standard deviation which are respectively the center position and the width of the Gaussian. Since the Gaussian can not be integrated mathematically, a lookup table of its cmf is available for a standard Gaussian of mean 0 and variance 1. Its typical PHI function allows the computation of the probability for any range of values.
  
•  Another important distribution is the uniform distribution, where each value has the same theoretical probability to occur. 
• For experiments purposes, it may be necessary to simulate noisy signals or environments. "Random Number Generators" are useful to this task.
  
• In many applications, the parameters of the distribution are estimated. The estimator is the method uses to estimate the value. The estimated value generally differ from the true value due to differents reasons: system calibration, operation noise, etc. The error is qualified in term of accuracy and precision.
• The accuracy refers to systematic errors, while precision refers to random errors. 

@uthor: Joan MOUBA 

Introduction To Digital Signal Processing

Digital Signal Processing (DSP) included mathematics, algorithms and methods use to manipulate signals.

Signals are generally data from the real-world, acquired through sensors. In the same manner that the ear captures sound waves and the eye captures electromagnetics waves with wavelengths between 400 and 700 nanometers to have a  representation in image and color of our environment.

A signal is represented as a function  of one or more parameters such as time (time domain),  range (spatial domain), frequency (spectral domain)... For example, we may represent a voltage, light intensity, sound pressure as a function of time, the rock intensity as a function of range.

The signal's shape can be visualized on a plot, and would give further information about

the signal's evolutions and trends (periodicity, statistics, ...).   

DSP has brought a revolutionary in science and engineering since 1970's when the computer becomes available. Today DSP impacts research and development in many fields such as communications, speech and sound processing, medical, geophysics,  oil and gas exploration and exploitation, military, radar, sonar, image processing, automobile, aeronautics, embedded systems ... DSP is used for example in noise reduction, speech generation and recognition, echo cancellation, artificial reverberation, computed tomography scanner,  ... DSP is present in many devices such mobile phones, pda, portable ECG analyzer, Mud logging, ... DSP is a crucial tool for any engineer working in high technology. 

@uthor: Joan MOUBA