Lecture 13 - FIR Filter Design and IIR Filter Stability (1).pptx

Lecture 13 - FIR Filter Design and IIR Filter Stability (1).pptx

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BMEN 3350 – Biomedical Component and System Design Lecture 13 FIR digital filter design, IIR filter stability, and introduction to IIR filter design.
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BMEN 3350 – Lecture 13 FIR vs. IIR Digital Filters Finite impulse response (FIR) filters rely only upon input values only, and are inherently stable. Infinite impulse response (IIR) filters rely on both input and output values (feedback), and have the potential to become unstable.
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BMEN 3350 – Lecture 13 FIR Filter Specifications Digital filter design typically begins with a frequency response specification. ω p – normalized cut-off frequency in the passband ω s – normalized cut-off frequency in the stopband δ 1 – maximum ripples in the passband δ 2 – minimum attenuation in the stopband [dB] Low-pass Filter Specification
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BMEN 3350 – Lecture 13 FIR Filter Specifications Digital filter design typically begins with a frequency response specification. ω p – normalized cut-off frequency in the passband ω s – normalized cut-off frequency in the stopband δ 1 – maximum ripples in the passband δ 2 – minimum attenuation in the stopband [dB] a p – maximum ripples in the passband a s – minimum attenuation in the stopband [dB] Low-pass Filter Specification
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BMEN 3350 – Lecture 13 FIR Filter Specifications High-pass Filter Specifications
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BMEN 3350 – Lecture 13 FIR Filter Specifications Band-pass Filter Specifications
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BMEN 3350 – Lecture 13 FIR Filter Specifications Band-stop Filter Specifications
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BMEN 3350 – Lecture 13 Frequency Normalization f s – sampling frequency f - frequency to normalize ω - normalized frequency
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BMEN 3350 – Lecture 13 Z-Transform of a Discrete Signal Recall that for a discrete-time signal, where And z is a complex variable of the form z = Ae
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BMEN 3350 – Lecture 13 Z-Transform of a Discrete Signal Consider the signal, x(n)={1,2,3,4,5,4,3,2,1} ; 0 ≤ n ≤ 8 Therefore:
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BMEN 3350 – Lecture 13 Relationship to Fourier Transform The Z-transform and Fourier transform are very similar: Since z is a complex variable of the form z = Ae , the Fourier transform is just a special case of the Z-transform where A = 1
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BMEN 3350 – Lecture 13 Fourier Transform of a Discrete Signal Fourier transform of x(n)={1,2,3,4,5,4,3,2,1} ; 0 ≤ n ≤ 8:
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BMEN 3350 – Lecture 13 Stability of FIR Filters The Z-transform of a given filter may be expressed as: The region of convergence (ROC) of a filter is given by the exterior of a circle in the complex plane which encompasses all of the poles of the transfer function. Since a FIR filter contains no poles by definition (i.e. ), it is inherently stable. We will revisit conditions of stability in more detail when we discuss IIR filters. b i - the feedforward filter coefficients (non-recursive part) a j - the feedback filter coefficients (recursive part) H 0 - a constant q i - the zeros of the transfer function p j - the poles of the transfer function
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BMEN 3350 – Lecture 13 FIR Filter Design The window method is frequently used to design FIR filters due to its simplicity of implementation. A window is a finite array consisting of coefficients selected to
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