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David Ernesto Troncoso Romero (National Institute of Astrophysics, Optics and Electronics (INAOE), Mexico) and Gordana Jovanovic Dolecek (National Institute of Astrophysics, Optics and Electronics (INAOE), Mexico)

Copyright: © 2015
|Pages: 11

DOI: 10.4018/978-1-4666-5888-2.ch151

Top## Introduction

A discrete-time signal is a sequence of numbers, so-called samples. A sample occurs at *T _{s}* seconds, then another sample appears during the next

A digital filter is a system that receives a discrete-time signal and returns another discrete-time signal with modified characteristics (Antoniou, 2006). A useful characteristic that many digital filters exhibit is the *Linear Time-Invariant* (LTI) property. When the input signal *x*(*nT _{s}*) has the value 1 for

When the values of the impulse response are real, the magnitude response is symmetric and the phase response is anti-symmetric around *f* = 0 (Saramaki, 1993). Thus, these responses are just observed in the range of values from *f* = 0 to *f* = 1/2*T _{s}*. Systems with real impulse response are used in a wide variety of applications, being one of the most common the traditional low-pass filter. This filter passes the frequency components of the input signal that range from 0 to

Complex Function: A function whose values are complex numbers.

Lineal Time-Invariant (LTI): Let us consider a filter that has as input a scaled sum of arbitrary discrete-time signals. If the output of this filter is equal to the scaled sum of the outputs of the filter to every one of these arbitrary signals, the filter is considered linear. If additionally the input signal is delayed in any possible amount and the output signal undergoes the same delay as the input, the filter is also considered time-invariant. A filter is LTI if it has both, linear and time-invariant properties.

Complex Number: A number composed by the sum of a real number a and an imaginary number i b , where b is a real number and i is defined as the square root of –1.

Multiplier: A component of a digital filter that takes the samples at its inputs and produces the multiplication of these values at its output.

Adder: A component of a digital filter that produces at its output the sum of the values of the samples present at its inputs.

Magnitude Response: A function of the frequency f where every value is obtained as the magnitude of the complex value of the frequency response in that frequency f . If the value of the frequency response in f is a complex number of the form a ( f ) + i b( f ), the magnitude of that number is given by {[ a ( f )] 2 + [ b ( f )] 2 } 1/2 .

Delay: A component of a digital filter that has one input and one output. The delay retains its input value during a sampling period T s and releases it at the next sampling interval.

Complex Exponential Signal: A signal whose samples are complex numbers, where the real and imaginary parts of the samples form, respectively, a cosine wave and a sine wave, both with the same frequency.

Phase Response: A function of the frequency f where every value is obtained as the phase of the complex value of the frequency response in that frequency f . If the value of the frequency response in f is a complex number of the form a ( f ) + i b( f ), the phase of that number is given by arctan { b ( f )/ a ( f )}.

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