《The Scientist and Engineer's Guide to Digital Signal Processing 》Study Noting

Chapter Two  Statistics,probability and noise

A signal is a description of how one parameter is related to another parameter.

For example, the most common type of signal in analog electronics is a voltage

that varies with time.   Since both parameters can assume a continuous range

of values, we will call this a continuous signal

Pay particular attention to the word: domain, a very widely used term in DSP.

For instance, a signal that uses time as the independent variable (i.e., the

parameter on the horizontal axis), is said to be in the time domain.  Another

common signal in DSP uses frequency as the independent variable, resulting in

the term, frequency domain

The variable, N, is widely used in DSP to represent the total number of

samples in a signal

Two notations for assigning sample numbers are commonly used.  In the first

notation, the sample indexes run from 1 to N  (e.g., 1 to 512).  In the second

notation, the sample indexes run from 0 to   (e.g., 0 to 511). N&1

Mathematicians often use the first method (1 to N), while those in DSP

commonly uses the second (0 to  ).  In this book, we will use the second N&1

notation.  Don't dismiss this as a trivial problem.  It will confuse you

sometime during your career.  Look out for it!

The mean, indicated by µ (a lower case Greek mu), is the statistician's  jargon

for the average value of a signal.  It is found just as you would expect: add all

of the samples together, and divide by N



The standard deviation is similar to the average deviation, except the

averaging is done with power instead of amplitude.  This is achieved by

squaring each of the deviations before taking the average (remember, power %voltage2).  To finish, the square root is taken to compensate for the initial

squaring.

为什么要进行傅里叶变换？原因何在

Fourier decomposition is important for three reasons. 

 First, a wide variety of signals are inherently created from superimposed sinusoids. 

 Audio signals are a good example of this.  Fourier decomposition provides a direct 

analysis of the information contained in these types of signals.  

Second, linear systems respond to sinusoids in a unique way: a sinusoidal input

always results in a sinusoidal output.  In this approach, systems are characterized

by how they change the amplitude and phase of sinusoids passing through them.  

Since an input signal can be decomposed into sinusoids, knowing how a system

 will react to sinusoids allows the output of the system to be found.  

Third, the Fourier decomposition is the basis for a  broad and powerful area of

 mathematics called Fourier analysis, and the even more advanced Laplace and z-transforms. 

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