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whose value is 1 over [0,1] and 0 elsewhere. The average of this standard uniform number x is
(1.1.2)
and its variance or deviation is
(1.1.3)
If you want another random number y with uniform distribution U(a,b), transform the standard uniform number x as follows:
(1.1.4)
For practice, we make a vector consisting of 1000 standard uniform numbers, transform it to make a vector of numbers with uniform distribution U(−1,+1), and then draw the histograms showing the shape of the distribution for the two uniform number vectors (Figure 1.5a1,a2) by running the following block of MATLAB statements.
Figure 1.5 Distribution (histogram) of noise generated by the rand()/ randn() command.
%nm01f05a.m N=1e4; x=rand(N,1); % An Nx1 noise vector with U(0,1) mx=mean(x); sgm2=sum((x-mx).̂2)/(N-1), var(x) subplot(221), M=20; hist(x,M) % Histogram having M=20 bins a=-1; b=1; y=(b-a)*x+a; %Eq.(1.1.4): 1000x1 noise vector with U(-1,1) subplot(222), hist(y,M) % Histogram
1.1.8.2 Random Number with Normal (Gaussian) Distribution
The numbers in a matrix generated by the MATLAB function ‘ randn(M,N)’ have normal (Gaussian) distribution with average m = 0 and variance σ2 = 1, as described by N(0,1). The random number x generated by ‘ rand()’ has the probability density function
(1.1.5)
If you want another Gaussian number y with a general normal distribution N(m,σ2), transform the standard Gaussian number x as follows:
(1.1.6)
The probability density function of the new Gaussian number generated by this transformation is obtained by substituting x = (y − m)/σ into Eq. (1.1.5) and dividing the result by the scale factor σ (which can be seen in dx = dy/σ) so that the integral of the density function over the whole interval ( ‐∞,+∞) amounts to 1.
(1.1.7)
%nm01f05b.m N=1e4; x=randn(N,1); % An Nx1 noise vector with N(0,1) subplot(223), M=20; hist(x,M) % Histogram having M=20 bins f=@(x,m,sgm)exp(-(x-m).̂2/2/sgm̂2)/sqrt(2*pi)/sgm; %Gaussian density ftn [Ns,Cs]=hist(x,20); dx=Cs(2)-Cs(1); % Bin width x_=[-5:0.01:5]; fx=f(x_,0,1); hold on, plot(x_,fx*dx*N,'r:') sgm=1/2; m=1; y=sgm*x+m; % An Nx1 noise vector with N(m,sgm̂2) subplot(224), hist(y,M) % Histogram [Ns,Cs]=hist(y,M); dy=Cs(2)-Cs(1); % Bin width y_=[-5:0.01:5]; fy=f(y_,m,sgm); hold on, plot(y_,fy*dy*N,'r:')
For practice, we make a vector consisting of 1000 standard Gaussian numbers, transform it to make a vector of numbers having normal distribution N(1,1/4), with mean m = 1 and variance σ2 = 1/4, and then draw the histograms for the two Gaussian number vectors (Figure 1.5b1,b2) by running the above block of MATLAB statements.
1.1.9 Flow Control
1.1.9.1 if‐end and switch‐case‐end Statements
An if‐ end block basically consists of an if statement, a sequel part and an end statement categorizing the block. An if statement, having a condition usually based on the relational/logical operator (Table 1.4), is used to control the program flow, i.e. to adjust the order in which statements are executed according to whether or not the condition is met, mostly depending on unpredictable situations. The sequel part consisting of one or more statements may contain else or elseif statements, possibly in a nested structure containing another if statement inside it. The switch‐case‐end block might replace a multiple if‐elseif‐..‐end statement in a neat manner.
Table 1.4 Relational operators and logical operators.
| Relational operator | Remark |
<
|
Less than |
<=
|
Less than or equal to |
==
|
Equal |
>
|
Greater than |
>=
|
Greater than or equal to |
∼=
|
Not equal (≠) |
&
|
and |
|
|
or |
∼
|
not |
Let us see the following examples:
1 (Ex. 1) A Simple if‐else‐end Block%nm119_1.m: example of if-end block t=0; if t>0 sgnt= 1; else sgnt= -1; end
2 (Ex. 2) A Simple if‐elseif‐end Block%nm119_2.m: example of if-elseif-end block if t>0 sgnt= 1 elseif t<0 sgnt= -1 end
3 (Ex. 3) An if‐elseif‐else‐end Block%nm119_3.m: example of if-elseif-else-end block if t>0, sgnt= 1 elseif t<0, sgnt= -1 else sgnt= 0 end
4 (Ex. 4) An if‐elseif‐elseif‐..‐else‐end Block%nm119_4.m: example of if-elseif-elseif-else-end block point= 85; if point>=90, grade= 'A' elseif point>=80, grade= 'B' elseif point>=70, grade= 'C' elseif point>=60, grade= 'D' else grade= 'F' end
5 (Ex.
