Lecture 3: Quantization Effects


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1 Lecture 3: Quantization Effects Reading: We have so far discussed the design of discretetime filters, not digital filters. To understand the characteristics of digital filters, we need first to understand the effects of quantization on both filter coefficients and signals. Most DSP systems are implemented using fixedpoint arithmetic, in which case the dynamic range of the data that can be represented is quite limited. Floatingpoint arithmetic helps alleviate this problem, but consumes too much power and costs more. Due to the very nature of DSP, where digital data are obtained through an A/D converter, floatingpoint precision is usually not required. We will solve the dynammic range problem by continuous scaling. Coefficient Quantization The strategy in designing a digital filter has been that we first design a discretetime filter with double floatingpoint precision, such as the use of Matlab, and then truncate (or round) the filter coefficients to implement the fixedpoint hardware. Number system: two s complement B xˆ X m b b i 0 i i X m x Bˆ where X m is the maximum value that x can take and x Bˆ b 0.b b b 3 b 4 b 5... b B. If b 0 0, 0 xˆ X m X m B, otherwise X m xˆ < 0. Define the step size to be the smallest quantity between numbers that can be represented by a quantizer, which is equal to X m B. Quantization error, e, is usually defined as xˆ x. Therefore for s complement truncation, the error is bounded by > e 0 rounding, the error is bounded by < e and for s complement 3
2 We also need to be concerned with overflow or underflow, which can usually be minimized by employing a saturation circuit after an addition or substraction. Effects of coefficient quantization on IIR systems: Hz ( ) M b k z k k N a k z k k N Az ( ) a k z k N ( p j z ) k j To analyze the sensitiviy of each pole position with respect to filter coefficients a k, we borrow the following chain rule from calculus: Az ( ) a z pi k p i Az ( ) z pi p i a k Az ( ) a p k N k i z pi p i then a k N Az ( ) p z pi i ( p i p j ) j, j i The overall sensitivity of each pole to quantization errors introduced by all coefficients can be approximated by p i N pi a. a k k k 3
3 Conclusion: Closely clustered poles are very sensitive to quantization error in filter coefficients (the same analysis applies to zeros as well). Solution: make sure that poles are NOT closely clustered! The use of ndorder section in filter design decouples the effects of quantization error on closely clustered poles. Let s take a look at implementing a pair of complex poles using a nd order filter. Hz ( ) ( γe jθ z )( γe jθ z ) γcosθz γ z γcosθ γ γcosθ and γ must be computed and rounded to the number of bits available. Suppose that we use a 4bit quantizer b 0.b b b 3. Both γcosθ and γ can take on the numbers from.000 to 0. ( to 0.875). Poles and zeros of a nd order filter can only occur at the intersection of the lines representing γcosθ and the semicircles representing γ. If γcosθ > 0.5, we can do two things. First, increase the number of bits before the binary point, or implement the factor of in γcosθ by a simple leftshift of bit after the coefficient multiplication, which involves some scaling consideration (will be discussed later). 33
4 Observations: Narrowband lowpass and narrowband highpass filters are most sensitive to coefficients quantization, which usually require more bits. Sampling at too high a rate is not necessarily good, pushing all poles closer to. Sensitivity increases for higherorder Direct Form realization. Poles may end up outside the unit circle after quantization and rounding. Cascade of ndorder sections is the common implementation to avoid instability. The stability of a ndorder section is guaranteed by ensuring γ to be less than. 34
5 Nonuniform density of poles and zeros of a ndorder section can be mitigated by using a coupled form structure. γcosθ γsinθ γsinθ γcosθ The quantized poles and zeros are at the intersections of evenly spaced horizontal and vertical lines. 35
6 Quantization effects on the FIR systems: Hz ( ˆ ) M [ hn ( ) hn ( )]z n 0 Hz ( ) Hz ( ) He ( jω ) M [ hn ( )]e jωn n 0 Frequency response distortion is linear with respect to the order of the filter. Quantization Effects on LinearPhase FIR Filters Quantized linearphase FIR filter is still linearphase. If zeros are closely spaced, implement small sections of zeros independently with a cascade form: Zeros on the unit circle: az  z . Zeros on the real axis: az  z . Zeros at and : z  and z . M ( hn ( ) ) e jωn ( B ) ( M ) n 0 Zeros of reciprocal conjugated pairs: ( γcosθz γ z ) γ ( γ γcosθz z ) 36
7 Signal Quantization Reading: 6.9 and Signal quantization involves truncation and rounding of data samples to certain precision, represented by the number of quantization bits per sample. A typical digital signal processing system has three sources of quantization noise. x(t) 6bit A/D converter 3bit DSP 6bit D/A converter y(t) The three sources of quantization noise can be represented by the following noise models: e (n) a k a k e (n) Impulse generator y(t) e 3 (n) e(n) is the difference between the exact sample value and the quantized value, due to the finiteword length of a quantizer. 37
8 Quantization noise is usually modeled as a random process, as the sample values cannot be predicted before hand. If there is no correlation between data samples, quantization noise can be approximated by a uniformly distributed probability density function, as shown below: PDF of e(n) e(n) _  Quantization noise mean 0. Quantization noise power  e. de Let s look at the first noise source, introduced by the A/D converter. The quantization noise power is expressed in terms of the square of quantizer step size. However, this quantity doesn t carry too much meaning if not compared with the signal power. This also means that we need to scale the signal so that the full dynamic range of the quantizer can be used. Assume a random input signal with variance σ (variance power  mean ) and zero mean. Take X m 4σ, which means that the dynamic range of the quantizer is set to be equal to 4 times of standard deviation σ of the input signal. We call this quantization strategy 4σ scaling. PDF of 4σ σ σ 4σ To calculate the signaltoquantization noise ratio, we first express signal power in terms of X m,asσ (X m /4)., the SNR can be calcu As the quantization noise power is lated as: ( X m B )
9 SNR B log B log db 6B.5dB 6 6 From the above equation, we know that we get 6dBgain in SNR per additional bit. For example, if B 6, SNR 89 db. But this formula is only true if 4σ scaling has been used. Q: What if a 5σ scaling strategy has been used? A: We still get 6 db per additional bit, but the constant term .5 db becomes 3.9 db. Q: If using a larger σscaling will give a lower SNR, why couldn t we always use a small σ scaling to get higher SNR? A: The equation is true only when the model of the noise is valid. If a smaller σ scaling strategy has been used, there will be a greater probability that the signal magnitude would exceed the dynamic range of the quantizer, causing highly nonlinear noise, which is not accounted for in the uniformly distributed noise model. Let s take a look at the third noise source, introduced by the D/A converter. Assuming that we quantize the internal data of 3 bits to 6 bits to be used by the D/A converter, the quantization noise can be approximated by a continuous random process uniformly distributed between half of the step size, similar to the case of A/D converter. This lowers the total SNR by another 3dB. (To calculate total SNR, all the noise powers are added up first, and then the ratio is calculated. 3dBsimply means that the signaltonoise power ratio is halved, as the noise power is doubled.) The second quantization noise source, introduced by rounding and truncation inside a DSP, is more complicated. First let s take any two numbers, a and b, of 6 bits each. If we multiply them together, ab will need at most 3 bits to represent the result without losing precision. 39
10 Quantization noise in IIR filters: 3 bits 6 bits IIR filters cannot be implemented with perfect precision. A quantizer must be placed at the output of a multiplier to limit the number of bits in a recursive calculation. We need to quantify this quantization noise in order to determine the number of bits needed to satisfy a given SNR. FIR filters, on the other hand, can be implemented without quantization noise, if the number of bits increases with the filter order. We seldom do this, however, as the A/D and D/A converters have already introduced some quantization noise. There is no point to enforce perfect precision inside a DSP when a nonzero noise power already exists. There are two effects to be considered when we design a quantizer for multipliers and accumulators:. Quantizer step size should be kept as small as possible because of the square term in quantization noise power.. X m should be large enough to prevent overflow. Of course these two effects contradict with each other and represent a design tradeoff, similar to the scaling strategy in determining the dynamic range of an A/D converter. 30
11 Example:Consider a direct form I secondorder section: Q: How many quantization noise sources are there in the filter? A: It depends on where quantizers are placed. We can have either or 5. Assuming that we have M noise sources, then the total noise power will be M To calculate the noise power after filtering, we need to borrow some math from random processes. We all know that the Fourier transform of a random process doesn t exist, but the Fourier transform of the power of a stationary random process does exist, called power spectrum density. The power spectrum density at the output of a filter is the multiplication of the power spectrum density of the input and the square of the filter frequency response, as shown in the following figure: H(e jω ) P x (e jω ) P y (e jω )P x (e jω ) H(e jω ) Therefore the noise power at the output of a filter is the integration of its power spectrum density: σ f π σ e He ( jω ) dω π π, 3
12 where we have assumed that the input noise power spectrum density is a constant variance σ (a white random process with zero mean). π By Parseval s Theorem,, then π He ( jω ) dω hn ( ) π n σ f σ e hn ( ). The output noise power can be calculated by either equation, as an integral of the power spectrum density over all frequencies, or as a scaled infinite summation of the impulse response squared. Usually it is easier to calculate the infinite summation, if the impulse response takes a closed form. Example:The noise power at the filter output is b Q X m B a Q a Example:The noise power at the filter output is σ f X m B hn ( ) X m B n. Q Q Q Q Q 3
13 s should always be less than. If s is greater than, then either x max is less than, implying that the previous stage didn t use the full dynamic range available, or the summation of the filter impulse response is less than, implying that the filter wasn t designed properly. Both cases point to a wasted dynamic range, reducing signal power unnecessarily. The design of the scal To calculate signal power, we use the same concept of power spectrum density, as the input signal is merely another random process. The signal power at the output of a filter is usually expressed as: π σ,whereσ s σ input He ( input is π jω ) dω σ input hn ( ) π n the input signal power, assuming that the input is also a white random process with zeromean. The total SNR is the ratio of the signal power to the noise power. But before we discuss SNR, we need to consider scaling, which connects the signal magnitude to the quantizer dynamic range. Scaling criterion #: Boundedinputboundedoutput. In this criterion we want to make sure that every node in the filter network is bounded by some number. If we follow the convention that each number represents a fraction (with a possible scaling factor), each node in the network must be constrained to have a magnitude less than to avoid overflow. Pick any node in the network w(n). Its response can be expressed as wn ( ) xn ( m)h w ( m) x max h w ( m) m m. To make sure that w(n) <, we need to introduce a scaling factor s such that x max s h w ( m) m. 33
14 ing factor is therefore to always give the signal the maximum dynamic range possible, as the quantization noise power is a constant once the step size is determined. Boundedinputboundedoutput criterion usually results in a very small s, which reduces the signal power and therefore the overall SNR. Criterion #: Frequency response criterion. In this criterion we input a narrowband signal x max cos ω o n to the filter. To avoid overflow at a node w(n) given this input signal, we need to insure that wn ( ) H w ( e jω o) x max. The scaling factor s should be chosen so that x max s max ( H w ( e jω ) ) 0 ω π Because max( H w ( e jω ) ) h w ( m), the scaling factor derived m using Criterion # is always smaller than the scaling factor derived using Criterion #. Example:Consider the following simple filter: b Q Q a Q: Assuming that x max, scale the input so that is always less than. 34
15 To find the scaling factor using Criterion #, we need to calculate the summation of the absolute values of the impulse response: s hm ( ) b a n m m a b. The output noise power is simply a To calculate signal power, assume that is a white random signal uniformly distributed between and . Its mean is 0 and its variance is /3. The signal power at the output of the filter is s b The total SNR is 3 a s. 3 b ( a ) s a b Several finiteprecision related problems: Signal quantization considerations for FFT. Signal quantization considerations for multiplications and additions. Normalized operations to mimic floatingpoint calculation. Complex operations such as divide and inverse usually use a table lookup to control the quantization effects. 35
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