Measuring and Controlling Load Power with Embedded Processors
Measuring and Controlling Load Power with Embedded Processors: Is There an Easy Way? Jim Gilbert Technical Fellow Covidien Energy-based Devices IEEE SPS – Denver May 10 th, 2011
Agenda • Introduction • What is so hard about measuring power? • Definitions • Uncertainty budgets for sensing power • Uncertainty in sensing power in the digital domain • Some digital implementation comparisons • Wrap up 2 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Quote of the Day “One person can use smoke and mirrors to make a demo that dazzles an audience. But shipping to a million customers will expose [a product’s] flaws and leave everyone looking bad… It is a cliché in our business that the first 90 percent of the work is easy, the second 90 percent wears you down, and the last 90 percent – the attention to detail – makes a good product. ” Ron Avitzur, “The Graphing Calculator Story, ” Pacific Technology, 2004. www. pacifict. com 3 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Introduction nwcommunityenergy. org The world market has become energy efficiency aware and mobile device dependent. This has affected many disparate markets from consumer and industrial lighting to arc welding and plasma cutters in some unique, and perhaps positive, but unexpected, ways… 4 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Introduction (continued) Consumer and industrial lighting examples: – Incandescent bulbs will no longer be found in the EU after 2012 and will be phased out of the US markets starting that same year; compact fluorescent [most with electronic ballast] is taking over [Kanter, NY Times, “European Ban …, ” Aug 31, 2009] – High-efficacy RF lamps, based in SMPS design techniques, have not only improved efficiency, but greatly extended lamp life and functionality, i. e. new applications and usage models [Knisley, EC&M, “RF Lighting …, ” Nov 1, 2002] – Industrial UV curing has combined SMPS design techniques with cathodeless RF lamps and LEDs to achieve incredible reliability, power density, and high factorythroughput rates [e. g. Fusion UV Systems’® Light Hammer® and Phoseon Technology’s SLM™] 5 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Introduction (continued) Arc welding and plasma cutter examples: www. torchmate. com – The jump from transformer-based to inverter-based arc welder technology has reduced costs and improved production quality in the highly demanding aerospace industry [Miller Electric Mfg. , “Aerospace Fabricator …, ” www. millerwelds. com] – Cathode spot control and arc plasma start/restrike phenomena require response speeds of 10 k. Hz or more bandwidth, which the new RF AC inverters with digital feedback control can offer over prior analog art [Weman, Welding Processes Handbook, 2003] – Scramjet plasma torches require active power control and benefit further in increased battery life from pulsed or sinusoidal power waveforms [Billingsely et al, “Improved Plasma …, ” Virginia Tech, AIAA Paper 2005 -3423] 6 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Introduction (continued) • The typical digital feedback controller requires some signal processing of the process variable sensors – This can be done in either the digital or analog realm, but in either case, this introduces a delay or latency into the loop that may affect control response – Each block in the control loop introduces both magnitude and phase errors, these affect accuracy and precision; this includes the signal processing – An overall budget for the accuracy and precision as well as the latency is necessary to architect the system appropriately 7 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
What is so hard about measuring power? • Huh? – Average AC real power measurements can be distorted by – Nonlinearities, HF limitations (>10 MHz), or noise – not in detail – Algorithmic approximations – yes, in detail – This talk will center primarily on uncertainties created by digital signal processing algorithmic approximations – These uncertainties can be minimized (or practically eliminated) by architectural design choices, i. e. sample frequency and averaging window selection – The signal processing typically contributes non-negligible latency that affects control loop dynamics – This talk will cover computational complexity as a relative measure – This talk will not go into detail about latency in control loop designs 8 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
What are we doing with power: Tracking Pavg • Power is a measure of the rate of work being done by the system ‒ Rate of heating, lifting, or, generally, energy conversion ‒ Tracking power allows control of the energy delivery • Need to track the envelope of the voltage and current waveforms • AM Power! • Amplitude Modulation is V(t) has its own bandwidth, often < c 9 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Tracking of Pavg (continued) • Which signal is being modulated: voltage or current? – Ideal voltage source → current is modulated by load fluctuations – Ideal current source → voltage is modulated by load fluctuations • If one or the other is well known and controlled, then you need only measure and control based on the other – simple • Sometimes, but… – Sources are not always “ideal” – e. g. matched to some maximum power transfer load; so, both voltage and current are modulated – The load in question, if it is interesting, may be non-linear; so, different, non-ideal spectra for voltage and current 10 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Tracking of Pavg: Non-linear Loads What does a non-linear load look like? i(t) • It could be an arc between a stick welder and its workpiece… • Here we see an ideal sinusoidal voltage source driving a spark gap that results in a highly non-linear current waveform • If distance is precisely known, then a voltage-source welder works here. But, that is why stick welders are currentsourced! 11 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver v(t)
Tracking of Pavg (continued) • And, just to make it more interesting, multiplication in the timedomain is convolution in the frequency-domain Voltage Baseband -Fv 0 Frequency * Current -Fi Fv 0 Assume voltage and current both have different amplitude modulation bandwidth Fi = Power - (Fv+Fi) 0 • Everything just got twice as hard! 12 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver (Fv+Fi)
Definitions and Refreshers • Instantaneous power – Voltage times current (note both are real valued signals) • Average Power definition – Pavg is defined as the average power absorbed by a load over some time interval, ∆t = (t 2 - t 1); (sign matters!) • For periodic signals – Average power of one period = average power of any number of periods, k – T = the period of v(t) and i(t) – k. T is integration time or averaging window “length” 13 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver Electrical Mechanical Fluid/Acoustic
Definitions (continued) S = Complex Power (of a singlefrequency sinusoid) |S| = Apparent Power (VA) Q = Reactive Power (VAR) P = Real Power (Watt) Complex Voltage and Current Vector (Phasor) Plot * Denotes complex conjugate Diagram from Wikipedia http: //en. wikipedia. org/wiki/AC_power 14 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Definitions (continued) • Accuracy and Precision – Accuracy is the relative difference between a reference and the measured values – Precision is the unbiased repeatability of the measurement – Accuracy is a metric for the non-random bias, or systematic error, and precision is a metric for the unbiased random error We will assume that we can calibrate out bias, or systematic error, and neglect covariance (by design), leaving primarily random error for this study: uncertainty error Diagram from Wikipedia http: //en. wikipedia. org/wiki/Accuracy_and_precision 15 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Definitions (continued) • Independent and random fractional uncertainties of products or ratios are added together in quadrature, i. e. by r. m. s. addition The system fractional uncertainty is The independent contributing fractional the uncertainty normalized by the uncertainties absolute value of the best estimate 16 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Definitions (continued): Big O • A function that defines an upper bound of another function ek f(x) O(ek) f(x) • In Computer Science Big O is used as a metric for computational complexity – MIPS/FLOPS are relative to a processor and can be misleading or confusing, when comparing algorithms versus specific implementations 17 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Definitions (continued) • We define computational complexity of an algorithm as: – Sum total of additions/subtractions and multiplies – Division is considered to be multiplication – Everything else not included (moving data) • Example: y[n]=A∙sum(x[n]), for N samples one multiply N-1 additions y[n] O(Nn) Order N, or , N operations for n samples 18 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Definitions (continued) • Sample rate conversion – handled similar to MIPS – Example: Direct (inefficient) form of decimation v[n] @ Fs MSPS N samples M Sum Pavg[k] @ Fs/M MSPS i[n] One sample, n, requires N computations One sample, k, requires N∙M computations Conversion: Multiply by M O(Nn) 19 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver O(NMk)
Uncertainty Budget Example: A/D Selection Let’s start apportioning an error budget from the analog side assuming digital is negligible (< 1/5 th)… • The voltage and current sensor chains each consist of a cascade of transfer functions with gain • The measured power is dependent on a linear combination of the voltage and current sensor gain product terms • Budget for the A/D and sensor contributions to total uncertainty error by r. m. s. addition Let’s say 15% ripple is OK in the total analog budget (pretty generous), and we split it evenly between the A/D’s and the sensors Then the A/D’s would be allowed 11% of the budget 20 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Uncertainty Budget Example (continued) Now we wish to calculate our A/D budget for finding the minimum number of bits of resolution • Starts by combining an equation for power with an equation for the propagation of uncertainty error • Requires finding the partial derivatives for each of the variable terms • But the tan() function goes to infinity! • Compromise: measure power precisely up to a PF of 0. 707, else, assume open circuit, or saturation Where, 21 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Uncertainty Budget Example (continued) • Substitute our earlier budgeted goal for the A/D chain (~11%) • Let’s say we only need 30% (or 2/3, or one standard deviation) margin for headroom (~7. 33%) • Allow for equal contributions in magnitude and phase errors from voltage and current chains • Remember δϕ implies <7° matched phase resolution including phase noise plus jitter • Not much dynamic range, but certainly low cost! Digital calculations must be < 1%! For this example one could use an 8 -bit A/D with 16 -bit math … and 3 d. B headroom/footroom 22 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Uncertainty in Calculating Average Power • When periodic signals are sinusoids, Squaring Pavg can be evaluated analytically to arrive at our earlier simplification, we’ll call “r. m. s. power” 23 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Average Power (continued) Well, the first problem is that the “r. m. s. power” simplification does not always hold… Pavg by integration • The average power by integration is true for all cases, while “r. m. s. power” is only an approximation for any case other than a load with a linear vi characteristic Pavg by multiplication of r. ms. values • Example: For second-harmonic with relative amplitude factors of xv and xi with reference to the fundamental the equations lead to noticeably different answers • Worse, yet, it is noisier! 24 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver Not equivalent for all xv and xi and noise is not “averaged out, ” i. e. the covariance assumption is violated
Deviation of power by r. m. s. multiplication 25 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Average Power: What else can go wrong? • Squaring a sinusoid: • Spectrum of sinusoid: -2 c 26 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver - c c 2 c
Average Power (continued) • This works beautifully because the average of a sinusoid across one or more of its periods is always = 0, for any k, ϕ, or t 1 • Integration can be approximated with a low-pass filter. Most obvious simplementation is a boxcar window => sinc() 27 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Average Power (continued) • Spectrum of Box Car Averaging Window with a coherently sampled signal Squared sinusoid Averaging Window (sinc) Only DC component remains after integration Pavg is a DC value. Perfect! 28 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Average Power (continued) • This perfect calculation even works with a distorted signal (harmonics)! 3 rd Harmonic 2 nd Harmonic Fundamental DC value reflects power of fundamental and harmonics 29 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
The Real World • What happens when the integration time or window is not a perfect integer multiple of the input period? – Pavg will vary or ripple over time – Why? 30 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Deviation of Average Power • Squared spectrum (with noise) and averaging filter • Output spectrum of wrong integration length wrt period Bummer 31 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Deviation of Average Power (continued) • Simulate and test Standard Deviation of Pavg with noise Variation of Pavg is defined by imperfect window With a perfect window, Pavg varies proportional to noise level 32 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Deviation of Average Power (continued) • But, this could get tedious and interpreting it gets complicated fast… 33 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Deviation of Average Power (continued) • And, results can vary and seem counter-intuitive • We need to understand better what is going on to optimize this! 34 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Discrete-time Analysis • Our approach in bounding this was to start in the continuous-time domain, then convert to the discrete-time domain • Simplifications – No harmonics (or noise) – Power Factor = 1, or PF angle = 0 * – Mathematically, normalized function x = voltage = current Common phase offset, not the PF angle • Pdiff, the difference in Pavg between an ideal integration window and a non-ideal integration window… 35 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Analysis (continued) • Pdiff( , T) at fixed c Pdiff( , c) at fixed T Note: offset by 0. 5 for illustration purposes. 36 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Analysis (cont) • Bounding Pdiff across all phase offsets, – Assuming is unknown, what is the worst case uncertainty of Pavg? – Find min and max of Pdiff – Define Pmax. Dev: Maximum deviation of Pdiff – Uncertainty of Pavg for stimulus signal x(t)=Asin( t+ ) 37 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Pmax. Dev for Continuous-time 38 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Continuous-Time to Discrete-Time Domain • Wow – this looks like the scalloping loss problem of FFT’s! • When input signal x(t) is digitized, uncertainties defined by Pmax. Dev (in the continuous-time domain) are transformed by the digitization process – Pavg is a combination of discrete and continuous variables – Phase offset θ and sinusoid frequency are continuous – Time is discrete – x(t) becomes x[n] at sample frequency Fs, Ts=1/Fs – T (the integration or averaging window) becomes NTs, where N is a positive integer – Pmax. Dev is affected by sampling the same way that the continuous-time Fourier Transform (CTFT) relates to the discrete-time Fourier Transform (DTFT) – Replicas every k. Fs, k {…, -2, -1, 0, 1, 2, …} 39 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Pmax. Dev for Discrete-time • This can be done for any integration method, but for Rectangular or boxcar integration: Pmax. Dev from continuous-time equation Shaping due to Rectangular integration Sampling in time at Ts creates multiple frequency copies at Fs Closed-form expression not known… yet. This result has been verified by simulation… 40 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
PRmax. Dev Simulation Figure • Nulls at m/N, m = 1, 2, 3…floor(N/2) Appears to fold over at Nyquist, , instead of sample frequency! 41 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Notes on Discrete Integration Methods • Discrete integration algorithms – Rectangular Method – Approximation with single points – Nulls cancel perfectly for coherent sampling with correct window length – Computationally, the simplest per sample – Trapezoidal Rule – Approximation with two point lines – More accurate (i. e. better sidelobe and stopband), but extra computations – Works better for non-coherent sampling or non-perfect window length – Simpson’s Rule – Approximation with three point polynomials – Most accurate for low/mid frequencies – Most computationally complex and noisy at high frequencies (near Nyquist) – Simpson’s Rule is NOT recommended 42 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Shaping Windows • Similar to a DFT, a shaping window can be placed in the signal processing chain to help reduce Pmax. Dev • This is the same as replacing the integrator with an FIR filter! 43 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Shaping Window (continued) • Pmax. Dev with shaping window and rectangular integration 44 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Now the Hard Part: Pulsed Stimuli • Example: sinusoid that is pulsed on/off, periodically • Assume on and off cycles are each an integer number of sinusoidal periods (this makes it much easier!) • Math really hard… we resorted to simulation to determine Pmax. Dev for any given scenario 45 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Pulsed Stimuli (continued) • Simulation results Fixed frequency Duty Cycle Variable 46 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver Duty Cycle Fixed
Summary of Pulsed Stimuli • Pulsed Stimuli – Pmax. Dev is null when the averaging (integration) window is an integer multiple of the pulse period – For Duty Cycles < 100%, the type of integration (Rectangular or Trapezoid) becomes less significant, i. e. a Rectangular window is about as good as it gets – But, shaping windows still help! 47 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Summary Thus Far… • We use the digital domain for noisy, non-linear loads • Eliminate Pmax. Dev by using boxcar averaging and – Choosing (or re-sampling) Fs to be an integer multiple of fc – Fs should be at least > 4 fc (due to Nyquist fold-over and squaring) – Choosing N (averaging window length) to be an integer multiple of the driving function period, i. e. NTs = Tc – Choose N to yield an integer multiple of pulsed stimuli period • If above not possible, minimize Pmax. Dev by – Choosing N such that Pmax. Dev is near a null – Using Trapezoidal integration instead of Rectangular – Or, using shaping window, or FIR filter for integrator – Nulls are in a different place compared to boxcar window 48 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Implementation Comparison Overview • Digital architectures for calculating Pavg – Wide-band, Mean-Squared § “Mean-VI” average power by integration § “RMS-PF” power by r. m. s. multiplication – Narrow-band, DFT § Goertzel DFT (single or multiple frequency) § Polyphase Demodulation (single or multiple frequency) – Comparisons § System resources § SNR performance via simulation 49 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Comparison Overview (continued) • Examining viable, efficient implementations – Others not included: under-sampling, sliding-DFT, IIR filters • Comparing apples-to-apples as best we can – System resources: spectrum, memory, computational complexity – Performance: accuracy and precision (STD), group delay (latency) • This study does not include – Calibration techniques – Parasitic or cable compensation techniques 50 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Average Real Power • Instantaneous Power • Average real power, general form Averaging filter, LPF Mean-VI Implementation • Mean-Square A good approximation for Pavg when inputs are sinusoidal is by multiplication of RMS Power Factor, PF RMS-PF Implementation – Harmonics must be equal between voltage and current (true, except non-linear) – Remember that noise is correlated by squaring, thus roughly doubles! 51 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Real Power by Fourier Transform (continued) • Complex numbers • Single-bin Fourier Transform of Voltage and Current • Real Power 52 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver Goertzel or Polyphase Demodulation Implementations
Quick Comparison • Architectures for calculating Pavg – Wide-band, Mean-Squared § Mean-VI § RMS-PF – Narrow-band, DFT § Goertzel § Polyphase Demodulation • Double spectrum (for squaring) • Power of fundamental and harmonics • Averaging filter notches-out squared harmonics • Normal spectrum (no squaring) • Power at single frequency only • DFT: § time length vs freq (bin) resolution § bin alignment … run multiple times for multiple frequencies and add them! 53 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Decimation can simplify – if you have the time • Once Pavg is averaged, it doesn’t need to run at the sampling rate, Fs, because it’s at baseband! – Therefore, can toss out samples or decimate – Keep every Mth sample, drop others. But, how much? • Before dropping samples, the signal must be bandlimited to the new Nyquist rate: = Control update rate …defined by control loop requirements Spectrum … π/M π 2π x[n] Drop Samples … π 2π 54 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver LPF M x[k]
Decimation (continued) • Combining Mean-VI and Decimation LPF Pavg v[n] @ Fs MSPS Decimation N samples Average LPF M Pavg[k] @ Fs/M MSPS i[n] Both LPFs …Combine! 55 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Decimation (continued) • Mean-VI and Decimation, nomenclature Averaging window of previous N samples No decimation Decimate by M k is every Mth sample of input (n) Averaging LPF and Decimation LPF are a combined function 56 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Decimation (continued) • Averaging window N, moving in time Sliding/overlap Window Domain of Polyphase Filter Structures Not recommended! 57 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver “Stepping”/non-overlap Window “Jumping” Window
Decimation (continued) • When a boxcar filter is used, M should be < N for effective filtering Example: N=8 Best 58 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Decimation (continued) • If the boxcar filter doesn’t cut it, add a designed FIR filter Filter coefficients other than 1 Wider main-lobe may help AM detection bandwidth of Pavg We added another multiply @ Fs. Computational complexity goes up. For M<N, use polyphase struct for efficiency. 59 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Mean-VI, Polyphase Moving Average • A polyphase moving average filter implementation • Wide-band average power calculation 60 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Polyphase Demodulation • Bandpass frequency content is demodulated to baseband complex value M=8 Normalized Freq (xπ) One phasor at ω = r(2π/M) yields one frequency bin LPF acts like a bandpass filter 61 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver h[n] is flexible
Polyphase Demod Implementation • A polyphase demodulation implementation • Narrow-band calculation only • Extract voltage and current AM waveforms before multiplying for Pavg – “Double” spectrum not needed anymore, but more computations since voltage and current are processed separately 62 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Goertzel Implementation • Goertzel: Single FFT bin via IIR filter (often used for tone detection) – Could be used for extracting phase information on RMS-PF – Calculate Pavg via complex values • Narrow-band calculation Goertzel operates at M≥N For M<N, use sliding-Goertzel 63 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Polyphase Demod and Goertzel • Carrier frequency fc, with respect to the DFT bin, is important, especially in performance comparisons Scalloping loss! 64 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver When Fs is an integer multiple of fc, DFT bins will align, otherwise, there may be scallop loss
RMS-PF Implementation • RMS-PF Implementation • Wide-band calculation 65 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
RMS-PF Implementation (continued) Boxcar Averaging Filter 66 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Computational Complexity Comparison Boxcar Filter RMS-PF Poly. D Process v(t) and i(t) individually Goertzel Mean-VI Shaped filter Must consider spectral content when comparing Big O! Typically desire to operate in M<N for lowest latency… Trade-off is higher STD (less processing gain) 67 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Conclusions • Best in show – Wide-band: Mean-VI Multistage § Minimal system resources § Most flexible filter design – Narrow-band: Polyphase Demodulation § Works best for negligible harmonic content § Efficient for M<N § Flexible filter design • Runners up – Narrow-band: Goertzel § Efficient for M>N, but filter inflexible – Wide-band: RMS-PF § SQRT and separate computation of phase are killers! 68 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Thank you Brian Roberts! I owe much to Brian Roberts for his help on everything, but especially on producing the simulations and the slides! 69 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Questions? 70 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
Supporting Slides 71 | Covidien Energy-based Devices | 5/10/2011 | IEEE SPS - Denver
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