I am quantizing an FIR filter's coefficients into 8-bit signed integers, which will be coded into hardware. How do I describe the errors in the quantization in my report? I only know of frequency response before and after quantization. TIA
Quantizing the filter coefficients, what are the criterias?
Started by ●June 8, 2006
Reply by ●June 8, 20062006-06-08
For FIRs, that's all there is. --Randy Mr. Ken wrote:> I am quantizing an FIR filter's coefficients into 8-bit signed integers, > which will be coded into hardware. How do I describe the errors in > the quantization in my report? I only know of frequency response > before and after quantization. > > TIA
Reply by ●June 8, 20062006-06-08
On Thu, 8 Jun 2006 11:24:53 +0800, "Mr. Ken" <Mr. Ken@asdf> wrote:>I am quantizing an FIR filter's coefficients into 8-bit signed integers, >which will be coded into hardware. How do I describe the errors in >the quantization in my report?Since the coefficients of a FIR filter are also its impulse response, the set of quantization errors can be thought of as the impulse response of an "error filter" that is added to the "true filter" to obtain the "quantized filter". Similarly, the Discrete Fourier Transform of the set of quantization errors can be thought of as an "error frequency response" that is added to the "true frequency response" to obtain the "quantized frequency response". You can even manipulate the quantization errors to obtain more desirable characteristics in this "error filter". Greg
Reply by ●June 8, 20062006-06-08
Greg Berchin wrote:> > Since the coefficients of a FIR filter are also its impulse response, > the set of quantization errors can be thought of as the impulse response > of an "error filter" that is added to the "true filter" to obtain the > "quantized filter". Similarly, the Discrete Fourier Transform of the > set of quantization errors can be thought of as an "error frequency > response" that is added to the "true frequency response" to obtain the > "quantized frequency response". You can even manipulate the > quantization errors to obtain more desirable characteristics in this > "error filter".one way to do that is with noise shaping the impulse response as it is input to the 8-bit quantizer. but we steer the noise from the stopband (where the absolute error in the FIR gain is more of a problem) into the passband where it is less of a problem. r b-j
Reply by ●June 8, 20062006-06-08
I am also working on a cpld project where I need to see the frequency responses of different coefficient wordlengths in order to obtain the optimum result as space requirements on the device are very tight. Are there any free utilities on the web to achieve this? Rick Mr. Ken wrote:> I am quantizing an FIR filter's coefficients into 8-bit signed integers, > which will be coded into hardware. How do I describe the errors in > the quantization in my report? I only know of frequency response > before and after quantization. > > TIA
Reply by ●June 8, 20062006-06-08
"realimag" <realimag@hotmail.com> wrote in message news:1149759795.627501.89100@h76g2000cwa.googlegroups.com...> I am also working on a cpld project where I need to see the frequency > responses of different coefficient wordlengths in order to obtain the > optimum result as space requirements on the device are very tight. Are > there any free utilities on the web to achieve this? > > Rick >If you use Xilinx FPGA in ISE, you may as well write some scripts to automate the entire procedure, then you can run fitting for all the wordlength. For Altera, I am not sure.
Reply by ●June 9, 20062006-06-09
> > one way to do that is with noise shaping the impulse response as it is > input to the 8-bit quantizer. but we steer the noise from the stopband > (where the absolute error in the FIR gain is more of a problem) into > the passband where it is less of a problem. >Sounds interesting...do you have any more details on how to perform this noise shaping, or a pointer to a paper/book?
Reply by ●June 9, 20062006-06-09
Rick, ONEoverT digital filter designer will let you see the responses of different coefficient wordlengths. you won't get any code output with the demo version however, but you probably don't need the full version anyway. Bob realimag wrote:> I am also working on a cpld project where I need to see the frequency > responses of different coefficient wordlengths in order to obtain the > optimum result as space requirements on the device are very tight. Are > there any free utilities on the web to achieve this? > > Rick > > Mr. Ken wrote: > > I am quantizing an FIR filter's coefficients into 8-bit signed integers, > > which will be coded into hardware. How do I describe the errors in > > the quantization in my report? I only know of frequency response > > before and after quantization. > > > > TIA
Reply by ●June 9, 20062006-06-09
in article 4489e9fe$0$1216$ba620e4c@news.skynet.be, NewLine at umts_remove_all_this@skynet.be wrote on 06/09/2006 17:37:>> >> one way to do that is with noise shaping the impulse response as it is >> input to the 8-bit quantizer. but we steer the noise from the stopband >> (where the absolute error in the FIR gain is more of a problem) into >> the passband where it is less of a problem. >> > > Sounds interesting...do you have any more details on how to perform this > noise shaping, or a pointer to a paper/book?quantizing: .------. x[n] ------>| Q{.} |---------> xq[n] '------' xq[n] = Q{ x[n] } = x[n] + e[n] Q{.} is the quantization operator (rounding or truncation) and e[n] =def xq[n] - x[n] = Q{x[n]} - x[n] is the quantization error (or "quantization noise"). it can be computed by taking the output of the quantizer and subtracting the input. what this does is model the quantizer as some kind of additive error source: .----- e[n] = xq[n] - x[n] | v x[n] ---------->(+)----------> xq[n] here, the transfer function from e[n] to xq[n] is Xq(z)/E(z) = 1 but e[n] is usually considered to be much smaller than x[n]. now if that is true (x[n] much larger than the round-off or truncation error of the quantizer), then the round-off error could be anything within -1/2 LSB and +1/2 LSB (and pretty random) and the q[n] error will be pretty white (up to Nyquist). so the spectrum of xq[n] is pretty much the same as x[n] except for some sorta flat white noise spectrum added in. if the spectrum of x[n] had some gaps in it where the energy got very low, then the whitish spectrum of e[n] fills in those gaps in the spectrum. noise shaping: .-------------. e1[n] .---| (z^-1)*G(z) |<----.----- e[n] = xq[n] - x1[n] | '-------------' | v - v x[n] ---->(+)-------------------->(+)-------> xq[n] + x1[n] notice that the output of the filter G(z) is subtracted from x[n] to result in x1[n] which is what is applied to the quantizer Q{.}. so the output now is xq[n] = Q{ x1[n] } = x1[n] + e[n] where x1[n] = x[n] - e1[n] and e1[n] = g[n] (*) e[n-1] (*) means discrete convolution also, the filter (z^-1)*G(z) must have at least one sample delay built in because it does not know the present roundoff error (the roundoff of the present sample hasn't happened yet) only the past roundoff errors, e[n-1], e[n-2], etc. G(z) can be any realizable (causal) filter and is actually the filter that is implemented (not (z^-1)*G(z)), but it only sees the past sample quantization errors, not the current quantization error. now the transfer function of quantization noise to output is Xq(z)/E(z) = 1 - (z^-1)*G(z) now G(z) has to be designed in such a way so that |Xq(z)/E(z)| will steer or "shape" the spectrum of the noise away from where you don't want it and put it where you don't care as much. the simplest noise shaper is one where G(z) = 1 so the noise shaping filter would be Xq(z)/E(z) = 1 - (z^-1) with frequency response: |Xq(e^jw)/E(e^jw)| = |1 - e^(-jw)| = 2*sin(w/2) and that is a high pass filter with no gain at DC at all. that means there is no noise at DC, little noise at low frequencies and increased quantization noise around Nyquist. now, for FIR filter design, what we are talking about is h[n] (with spectrum H(z) where z = e^jw) instead of x[n] and the quantized impulse response is hq[n] with spectrum Hq(z). if we look at the gain of the FIR filter with a "dB lens", we will care more about errors in the spectrum Hq(e^jw) at frequencies where |H(e^jw)| is small (resulting in bigger dB error) than where |H(e^jw)| is large. so we want |Hq(e^jw)/E(e^jw)| = |1 - (e^-jw)*G(e^jw)| proportional to |H(e^jw)| or |1 - (e^-jw)*G(e^jw)|^2 = A*|H(e^jw)|^2 for some real constant A>0 . so given the "ideal" FIR, h[n]. with spectrum H(e^jw), you want to design G(z) so that the above equation is as accurate as possible, then run your h[n] through the noise-shaped quantizer: .-------------. e1[n] .---| (z^-1)*G(z) |<----.----- e[n] = hq[n] - h1[n] | '-------------' | v - v h[n] ---->(+)-------------------->(+)-------> hq[n] + h1[n] or .--------. e1[n] .---| G(z) |<----- e[n-1] = hq[n-1] - h1[n-1] | '--------' v - .------. h[n] ---->(+)----------------->| Q{.} |------> hq[n] = Q{h1[n]} + h1[n] '------' and see what you get out: hq[n]. does that make sense. otherwise, all i can recommend is Google. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●June 10, 20062006-06-10
in article C0AF954C.153EA%rbj@audioimagination.com, robert bristow-johnson at rbj@audioimagination.com wrote on 06/09/2006 21:14:> otherwise, all i can recommend is Google.Google: quantized FIR coefficients (no quotes) and you get a lot of nice hits including http://www.ece.vt.edu/fac_support/dspcl/docs/SPMag03.pdf and http://hjem.get2net.dk/jjn/quantfir.htm http://hjem.get2net.dk/jjn/download/quantfir.pdf the latter pair approaches this problem from the noise shaping POV as i had just previously. probably he does a better job. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
