Everything You Ever Wanted to Know About Filters* Class 5: Digital Filters III: Finite Impulse Response Filters / Conclusion June 12, 2015 Charles J. Lord, PE President, Consultant, Trainer Blue Ridge Advanced Design and Automation * But were afraid to ask

This Week’s Agenda 6/8 Analog Filters I: Resonant Circuits and Passive Filters 6/9 Analog Filters II: Active Filters 6/10 Digital Filters I: Sampling and the Z-Transform 6/11 Digital Filters II: Infinite Impulse Response Filters 6/12 Digital Filters III: Finite Impulse Response Filters and Conclusion

This Week’s Agenda 6/8 Analog Filters I: Resonant Circuits and Passive Filters 6/9 Analog Filters II: Active Filters 6/10 Digital Filters I: Sampling and the Z-Transform 6/11 Digital Filters II: Infinite impulse response filters 6/12 Digital Filters III: Finite Impulse Response Filters and Conclusion

Why the FIR? Amongst all the obvious advantages that digital filters offer, the FIR filter can guarantee linear phase characteristics. Neither analogue or IIR filters can achieve this. There are many commercially available software packages for filter design. However, without basic theoretical knowledge of the FIR filter, it will be difficult to use them.

Finite Impulse Response Filter

Properties of an FIR Filter Filter coefficients: x[n] represents the filter input, bk represents the filter coefficients, y[n] represents the filter output, N is the number of filter coefficients (order of the filter).

Properties of an FIR Filter Filter coefficients: FIR equation Filter structure

Properties of an FIR Filter If the signal x[n] is replaced by an impulse [n] then:

Properties of an FIR Filter Finally: where The coefficients of a FIR filter are the same as the impulse response samples of the filter!

Frequency Response of an FIR Filter By taking the z-transform of h[n] H(z): Replacing z by ej in order to find the frequency response leads to:

Frequency Response of an FIR Filter Since e-j2k = 1 then: Therefore: FIR filters have a periodic frequency response and the period is 2.

Phase Linearity of an FIR Filter A causal FIR filter whose impulse response is symmetrical is guaranteed to have a linear phase response. Even symmetry Odd symmetry

Design Procedure To fully design and implement a filter five steps are required: (1) Filter specification. (2) Coefficient calculation. (3) Structure selection. (4) Simulation (optional). (5) Implementation.

Filter Specification - Step 1

Coefficient Calculation - Step 2 There are several different methods available, the most popular are: Window method. Frequency sampling. Parks-McClellan. We will just consider the window method.

Window Method First stage of this method is to calculate the coefficients of the ideal filter. This is calculated as follows:

Window Method Second stage of this method is to select a window function based on the passband or attenuation specifications, then determine the filter length based on the required width of the transition band.

Window Method The third stage is to calculate the set of truncated or windowed impulse response coefficients, h[n]: for Where: for

Designing in the Real World Matlab has been a tool of choice for many for many years for designing FIR and IIR filters Like we saw for analog filters, there are an increasing amount of vendor-supplied tools, particularly for using ‘DSP enabled’ microcontrollers For those of us in the ARM world, we return to CMSIS…

CMSIS and CMSIS-DSP

Filters in CMSIS DSP • Biquad Cascade IIR Filters Using Direct Form I Structure • Biquad Cascade IIR Filters Using a Direct Form II Transposed Structure • High Precision Q31 Biquad Cascade Filter • Convolution • Partial Convolution • Correlation • Finite Impulse Response (FIR) Decimator • Finite Impulse Response (FIR) Filters • Finite Impulse Response (FIR) Lattice Filters • Finite Impulse Response (FIR) Sparse Filters • Infinite Impulse Response (IIR) Lattice Filters • Least Mean Square (LMS) Filters • Normalized LMS Filters • Finite Impulse Response (FIR) Interpolator

But WAIT – There’s more… Upcoming blog on integrating digital filters on ARM Posted ‘hands-on’ project(s) on using CMSIS and K64 Stay tuned to Design News website as well as my company (see last slide)

Filters – a Review Passive filters: Active Filters: simple to implement require no power supplies Have some finite Z Active Filters: Require power Subject to clipping, saturation, and amp BW More components and cost Allow amplification

Filters – part 2 Digital Filters Ideal if the signals are already digitized (sampled) Allows for additional effects (warping, etc) No additional parts or real estate * Provide finite time delay Aliasing

This Week’s Agenda 6/8 Analog Filters I: Resonant Circuits and Passive Filters 6/9 Analog Filters II: Active Filters 6/10 Digital Filters I: Sampling and the Z-Transform 6/11 Digital Filters II: Infinite impulse response filters 6/12 Digital Filters III: Finite impulse response filters and Conclusion

Please stick around as I answer your questions! Please give me a moment to scroll back through the chat window to find your questions I will stay on chat as long as it takes to answer! I am available to answer simple questions or to consult (or offer in-house training for your company) c.j.lord@ieee.org http://www.blueridgetechnc.com http://www.linkedin.com/in/charleslord Twitter: @charleslord