Significant Figures
Signicant gures of a number are those that can be used
with condence.
Rules for identiing sig. figures:
All non-zero digits are considered significant. For example, 91
has two significant digits (9 and 1), while 123.45 has five
signifi
_1.6487211.o
_ 1.648721
V Using one term: 9 =1 6:, 100%: 39.3
1/ Using two terms:
eU-5=1+0.5=1.5 21'648721_1'5100%=9.02% s 21'5_1'0100%=33.3%
* 1.648721 1.5
1" Using three terms:
2 _ _
e05 =1+0.5+=1.525 =w10m=14m 7 =M10m=76904
2! 1.648721 1.625
51 11.0"
Prespecified Error
- We can relate (as) to the number of significant
figures in the approximation,
50, we can assure that the result is correct to at
least n significant figures if the following criteria
is met:
as =(0.5 x102_") %
See Example 3.2 p56
Approximate Error
The true error is known only when we deal with functions that
can be solved analytically.
In many applications, a prior true value is rarely available.
For this situation, an alternative is to calculate an
approximation of the error usin
Round-off Errors
- Numbers such as 11:, e, or \H cannot be expressed by
a fixed number of significant figures.
- Computers use a base-2 representation, they cannot
precisely represent certain exact base-10 numbers
- Fractional quantities are typically rep
function I = cplxcomp(p1,p2)
% I = cplxcomp(p1,p2)
% Compares two complex pairs which contain the same scalar elements
% but (possibly) at differrent indices. This routine should be
% used after CPLXPAIR routine for rearranging pole vector and its
% corre
function [b,a] = ladr2dir(K,C)
% Lattice/Ladder form to IIR Direct form Conversion
% -% [b,a] = ladr2dir(K,C)
% b = numerator polynomial coefficients
% a = denominator polymonial coefficients
% K = Lattice coefficients (reflection coefficients)
% C = Ladd
Approximations and Round-Off Errors
For many engineering problems, we cannot obtain analytical
solutions.
Numerical methods yield approximate results, results that are
close to the exact analytical solution.
Only rarely given data are exact, since they o
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Professors are having a hard time monitoring whether
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SOLUTION
Using barcode attached to the ID c
Example
The exponential function can be computed using Maclaurin
series as follows: 2 3
x x x
9 21+): +-+
2! 3! 1?!
Estimate e"-5 using series, add terms until the absolute value of
approximate error so fall below a pre-specified error 8;
conforming with
Approximations and Errors
- The major advantage of numerical analysis is that
a numerical answer can be obtained even when a
problem has no analytical solution.
- Although the numerical technique yielded close
estimates to the exact analytical solutions,
Example 3.1
(h) The percent relative error for the bridge is [Eq. (3.3)]
I
Er Tmimib =0.01%
and for the rivet it is
l
= % = a)
a, 101m 10
Thus, although both measurements have an error of 1 cm. the relative error for the rivet is
much greater. We would co
Significant Figures
Scientific Notation
- If it is not clear how many, if any, of zeros are significant. This
problem can be solved by using the scientific notation
0.0013 = 1.3*10-3 0.00130 = 1.30"10-3
2 sig. figures 3 sig. figures
- If a number is expre
function [b,a] = cas2dir(b0,B,A);
% CASCADE-to-DIRECT form conversion
% -% [b,a] = cas2dir(b0,B,A)
% b = numerator polynomial coefficients of DIRECT form
% a = denominator polynomial coefficients of DIRECT form
% b0 = gain coefficient
% B = K by 3 matrix
Accuracy and Precision
Increasing accuracy
- Accuracy refers to how
closely a computed or
measured value agrees
with the true value.
- Precision refers to how
closely individual
computed or measured
values agree with each
other.
- Bias refers to systemati
function [Hr,w,a,L] = Hr_Type1(h);
% Computes Amplitude response Hr(w) of a Type-1 LP FIR filter
% -% [Hr,w,a,L] = Hr_Type1(h)
% Hr = Amplitude Response
% w = 500 frequencies between [0 pi] over which Hr is computed
% a = Type-1 LP filter coefficients
% L
True Error
u True error (E!)
True. error (E,) or Exact value of error
= true value approximated value
I:I True percent relative error ( 8: )
True error
True percent relative error : E: : X 100 (941)
True value
_ true value approximated value
_ x 100 (94
Truncation Errors
- Truncation errors are those that result using
approximation in place of an exact mathematical
procedure.
V0341) V(z;. )
1+1 1
w~w_
dAt
function [b,a] = par2dir(C,B,A);
% PARALLEL-to-DIRECT form conversion
% -% [b,a] = par2dir(C,B,A)
% b = numerator polynomial coefficients of DIRECT form
% a = denominator polynomial coefficients of DIRECT form
% C = Polynomial part of PARALLEL form
% B =
Approximate Error
- In many numerical methods a present approximation is
calculated using previous approximation:
Note:
-The sign ofg or 8: may be positive or negative
- We interestead in whether the absolute value is lower
than a prespecified toleran
Error Definition
Numerical errors arise from the use of approximations
Truncation errors Round-off errors
Result when Result when numbers
approximations are used having limited significant
to represent exact figures are used to
mathematical procedure. r
Example 3.1
Calculation of Errors
Problem Statement. Suppose that you have the task of measuring the lengths of a bridge
and a rivet and come up with 9999 and 9 cm. respectively. If the true values are 10,000 and
ID cm. respectively. compute (a) the true
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Batangas State University
College of Engineering, Architecture and Fine Arts
ECE/ICE/MexE Department
ECE 415
Signals, Spectra and Signal Processing
Laboratory Report No.3
Frequency Analysis
Submitted to:
Engr.Anton Louise P. De Ocampo
Submitted by:
Mandig
Engineering is a broad term that covers a wide range of applications and industries. Combining
mathematics, science and technology, engineers produce creative solutions to real world problems. As a
result there are many different types of engineering degr
function [K,C] = dir2ladr(b,a)
% IIR Direct form to pole-zero Lattice/Ladder form Conversion
% -% [K,C] = dir2ladr(b,a)
% K = Lattice coefficients (reflection coefficients), [K1,.,KN]
% C = Ladder Coefficients, [C0,.,CN]
% b = Numerator polynomial coeffic
BATANGAS STATE UNIVERSITY
College of Engineering, Architecture and Fine Arts
ECE/ICE/MEXE Department
ECE 409 - Digital Principles and Logic Design
Laboratory Report No. 9
Sequential Circuits
Submitted By:
Bernardo, McAlvin P.
ECE 4102
Submitted to:
Engr.
Batangas State University
College of Engineering, Architecture, and Fine Arts
ECE/ICE/MEXE Department
ECE 409
Digital Principles and Logic Design
Experiment No. 3
Simplification of Boolean Functions
Submitted By:
Bernardo, McAlvin P.
ECE 4102
Submitted To
Numerical analysis has numerous applications in all fields of science and some fields of engineering,
and essentially any type of work that requires calculations to give very precise solutions.
The point of numerical analysis is to analyze methods that ar