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UW-Madison ECE 734 - The Swiss Army Knife of Digital Networks

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The Swiss Army Knife of Digital NetworksReferencesThe Swiss Army Knife of Digital Networks ___________________ by Richard Lyons and Amy Bell ___________________ This article describes a general discrete-signal network that appears, in various forms, inside many DSP applications. So the "DSP Tip" for this column is for every DSP engineer to become acquainted with this network. Figure 1 shows how the network's structure has the distinct look of a digital filter—a comb filter followed by a 2nd-order recursive network. However, we do not call this unique general network a filter because its capabilities extend far beyond simple filtering. Through a series of examples, we illustrate the fundamental strength of the network: its ability to be reconfigured to perform a surprisingly large number of useful functions based on the values of its seven control parameters. z -1z -1z -Ny(n)x(n)c1a2b2b1b0a1a0Comb 2nd-order recursive network (biquad)-+ Figure 1. General discrete-signal processing network. The general network has a transfer function of H(z) = (1 -c1z-N) b0 + b1z-1 + b2z-2 1/a0 -a1z-1 -a2z-2 . (1) From here out, we'll use DSP filter lingo and call the 2nd-order recursive network a "biquad" because its transfer function is the ratio of two quadratic polynomials. The tables in this article list various signal processing functions performed by the network based on the an, bn, and c1 coefficients. Variable N is the order of the comb filter. Included in the tables are depictions of the network's impulse response, z-plane pole/zero locations, as well as frequency-domain magnitude and phase responses. The frequency axis in those tables is normalized such that a value of 0.5 represents a frequency of fs/2 where fs is the sample rate in Hz. Moving Averager: Referring to the first entry in Table 1, this network configuration is a computationally-efficient method for computing the N-point moving average of x(n). Also called a recursive running sum, or boxcar averager, this structure is equivalent to an N-tap direct convolution FIR filter with all the coefficients having a value of 1/N. However, this moving averager is efficient because it performs only one add and one subtract per output sample regardless of the value of N. (Whereas an N-tap direct convolution FIR filter must perform N-1 additions per output sample.) The moving averager's transfer function is Hma(z) = (1/N)(1-z-N)/(1-z-1). Hma(z)'s numerator results in N zeros equally spaced around the z-plane's unit circle located at z(k) = ej2πk/N, where integer k is 0≤k<N. Hma(z)'s denominator places a single pole at z = 1 on the unit circle, canceling the zero at that location. 1Table 1. General Functions. Function and coefficients Network behavior impulse response z-plane magnitude (dB) phase (rad.) Moving Averager a0 = 1, a1 = 1, a2 = 0, b0 = 1/N, b1 = 0, b2 = 0, c1 = 1, N = 8 0510-0.100.10.2Time-101-101Real partImaginary part-0.500.5-20-100-0.500.5-5051/NFrequency Frequency Differencer a0 = 1, a1 = 0, a2 = 0, b0 = 1, b1 = -1, b2 = 0, c1 = 0 0510-101Time-101-101Real partImaginary part-0.500.5-20-100-0.500.5-202Frequency Frequency Integrator a0 = 1, a1 = 1, a2 = 0, b0 = 1, b1 = 0, b2 = 0, c1 = 0 0.500.5-202051000.51Time-101-101Real partImaginary part-0.500.5-20-100Frequency Frequency Leaky Integrator a0 = 1, a1 = 1-α, a2 = 0, b0 = α, b1 = 0, b2 = 0, c1 = 0, α = 0.1 -1 0 1-101Real partImaginary part-0.5 0 0.5-20-100-0.5 0 0.5-202α = 0.5α = 0.1α = 0.5α = 0.10 10 2000.10.2Timeα = 0.1Frequency Frequency 1st-order Delay Network a0 = 1, a1 = -R, a2 = 0, b0 = R, b1 = 1, b2 = 0, c1 = 0 -1 0 1-101Real partImaginary part-0.5 0 0.500 5 1000.51Timezero atz = 11delay∆= 0.2-0.5 0 0.511.11.2Group delay0.2/fsR = 0.91delay∆= 0.2Frequency Frequency 2nd-order Delay Network a0 = 1, a1 = -R1, a2 = -R2, b0 = R2, b1 = R1, b2 = 1, c1 = 0 -1 0 1-101Real partImaginary part-0.5 0 0.50zerosnotshowndelay∆ = 0.3-0.5 0 0.51.522.50.3/fsGroup delay0 5 1000.51Time2.3R1 = -0.182R2 = 0.028delay∆ = 0.3FrequencyFrequency Differencer: This is a discrete version of a 1st-order differentiator. An ideal differentiator has a frequency magnitude response that's a linear function of frequency, and this network only approaches that ideal at low frequencies relative to fs. Integrator: This structure performs the running summation of the x(n) inputs samples, making it the discrete-time equivalent of a continuous-time integrator. Leaky Integrator: This network configuration, also called an exponential averager, is a venerable structure used in lowpass filter implementations for random noise reduction. It is a 1st-order IIR filter where, for stable lowpass operation, the constant α lies in the range 0<α<1. This nonlinear-phase filter has a single pole at z = 1-α on the z-plane, and a transfer function of Hli(z) = α/[1-(1-α)z-1]. Small values for α yield narrow passbands at the expense of increased filter response time. Table 1 shows the filter's behavior for α = 0.1 as solid curves. For comparison, the frequency domain performance for α = 0.5 is indicated by the dashed curves. 21st-order Delay Network: A subclass of a 1st-order IIR Filter, the coefficients in Table 1 yield an allpass network having a relatively constant group delay at low frequencies. The network's delay is Dtotal = 1 + ∆delay samples where ∆delay, typically in the range of -0.5 to 0.5, is a fraction of the 1/fs sample period. For example, when ∆delay is 0.2, the network delay (at low frequencies) is 1.2 samples. The real-valued R coefficient is R = -∆delay ∆delay + 2 (2) producing a z-plane transfer function of H1,del(z) = (R+z-1)/(1+Rz-1) with a pole at z = -R and a zero at z = -1/R. Performance for ∆delay = 0.2 (R = 0.91) is shown in Table 1 where we see the magnitude response being constant. The band, centered at DC, over which the group delay varies no more than |∆delay|/10 from the specified Dtotal value, the bar in the group delay plot, ranges roughly from 0.1fs to 0.2fs for 1st-order networks. So if your signal is oversampled, making it low in frequency relative to fs, this 1st-order allpass delay network may be of some use. If you propose its use in a new design, you can impress your colleagues by saying this network is based on the Thiran Approximation [1]. 2nd-order Delay Network: A subclass of a 2nd-order IIR Filter, the coefficients in Table 1 yield an allpass network having a relatively constant group at low frequencies.


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UW-Madison ECE 734 - The Swiss Army Knife of Digital Networks

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