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lab5a - Purdue University ECE438 Digital Signal Processing...

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Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 5: Digital Filter Design (Week 1) October 6, 2010 1 Introduction Hello, This is the first part of a two week laboratory in digital filter design. The first week of the laboratory covers some basic examples of FIR and IIR filters, and then introduces the concepts of FIR filter design. Then the second week covers systematic methods of designing both FIR and IIR filters. 2 Background on Digital Filters In digital signal processing applications, it is often necessary to change the relative ampli- tudes of frequency components or remove undesired frequencies of a signal. This process is called filtering . Digital filters are used in a variety of applications. In Laboratory 4, we saw that digital filters may be used in systems that perform interpolation and decimation on discrete-time signals. Digital filters are also used in audio systems that allow the listener to adjust the bass (low-frequency energy) and the treble (high frequency energy) of audio signals. Digital filter design requires the use of both frequency domain and time domain tech- niques. This is because filter design specifications are often given in the frequency domain, but filters are usually implemented in the time domain with a difference equation. Typi- cally, frequency domain analysis is done using the Z-transform and the discrete-time Fourier Transform (DTFT). In general, a linear and time-invariant causal digital filter with input x ( n ) and output y ( n ) may be specified by its difference equation y ( n ) = N 1 summationdisplay i =0 b i x ( n i ) M summationdisplay k =1 a k y ( n k ) (1) Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494- 0340; [email protected]
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Purdue University: ECE438 - Digital Signal Processing with Applications 2 where b i and a k are coefficients which parameterize the filter. This filter is said to have N zeros and M poles. Each new value of the output signal, y ( n ), is determined by past values of the output, and by present and past values of the input. The impulse response, h ( n ), is the response of the filter to an input of δ ( n ), and is therefore the solution to the recursive difference equation h ( n ) = N 1 summationdisplay i =0 b i δ ( n i ) M summationdisplay k =1 a k h ( n k ) . (2) There are two general classes of digital filters: infinite impulse response (IIR) and finite impulse response (FIR). The FIR case occurs when a k = 0, for all k . Such a filter is said to have no poles, only zeros. In this case, the difference equation (2) becomes h ( n ) = N 1 summationdisplay i =0 b i δ ( n i ) . (3) Since (3) is no longer recursive, the impulse response has finite duration N .
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