# Chapter 3 discrete fourier transform 76 fir filter

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Unformatted text preview: 0 \$  " ¢0 § ¥ ¡ 10¢¡¡ 0  " ¢0 § ¥ ¨§¡  §¡ ¡ ¡ § This is the impulse response of the continues system with amplitude response . We need to discretized the previous expression to obtain the impulse response of a discrete system: © ¥  © ¤ (3.40) ¥§¡  § ¢ §   ¢ £ §¢ the factor comes from equation (1.44); this is a scaling factor that allows us to say that . the Fourier transform of the discrete and continuous signals are equal in 3  § ¢ q) § ¢ 83 &E The ﬁnal expression of the digital ﬁlter is given by (3.41) 3 ¢ ¡ " 44" ¨0D#IAD) B I) 1 I0¦ I) 44" §  ) ¥ § ¢  ¢ ¡0 \$  " ¢0 § £§   "" ¦) 1)B) ' E E) E "" ¡ It is clear that this is a IIR ﬁlter (inﬁnite impulse response ﬁlter). A FIR ﬁlter is obtained by truncating the IIR ﬁlter: (3.42) 3 ¦ ¢ ¡ " ¨ 44" ¨0D#IAD) B I) 1 I0¦ I) 44" ©E §  ) ¥ § § ¢ ¡0 \$  " ¢0 § £§   "" ¦) 1)B) ' E E) E "" ¨ ¡ In this case we have a ﬁlter of length . When the ﬁlter is truncated the actual amplitude spectrum of the ﬁlter is not equal to the desired or ideal amplitude spectrum (3.35). This point has already been studied in section (1.2.3) where we examined the spectral artifacts that are introduced when a signal is truncated in time. In Figure for ﬁl(3.4) we display the impulse response of a ﬁlter of cut-off frequency ter lengths ( ) , and . We also display...
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## This note was uploaded on 11/29/2012 for the course GEOPHYSICS 426 taught by Professor Sacchi during the Winter '12 term at University of Alberta.

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