1 Digital Signal Processing Quantization Dr. Dung Trung Vo Telecommunication Divisions Department of Electrical and Electronics August, 2013 Quantization Process Analog to digital conversion: Sampling and quantization are necessary for any digital signal processing operation on analog signals: Samplehold and ADC may be separate modules or may reside on board the same chip CuuDuongThanCong.com https:fb.comtailieudientucntt2 Signal quantization: Fullscale range R: in practice are between 1–10 volts. Quantization width or the quantizer resolution: Number of quantization levels: 2B R B Q 2 AD converter Bipolar ADC: Unipolar ADC: Rounding: Truncation: ADC Classification xQ(nT) kQ 2 ( ) 2 Q kQ Q x nT kQ xQ(nT) kQ kQ x(nT) kQQ 2 ( ) 2 R R x nT Q x nT R 0 Q( ) CuuDuongThanCong.com https:fb.comtailieudientucntt3 Quantization error: Error distribution: Mean: Variance: Rootmeansquare (rms) error: Ratio of R and Q is a signaltonoise ratio (SNR, dynamic range of the quantizer): Quantization Process e(nT) xQ(nT) x(nT) 1 2 0 2 _ Q Q ede Q e 12 1 2 2 2 2 _ 2 Q e de Q e Q Q 12 _ 2 Q e e rms B R Q SNR 20log10 6 Example: In a digital audio application, the signal is sampled at a rate of 44 kHz and each sample quantized using an AD converter having a fullscale range of 10 volts. Determine the number of bits B if the rms quantization error must be kept below 50 microvolts. Then, determine the actual rms error and the bit rate in bits per second Quantization Process CuuDuongThanCong.com https:fb.comtailieudientucntt4 Probabilistic interpretation of the quantization noise: quantized signal xQ(n) as a noisy version of the original unquantized signal x(n) to which a noise component e(n) has been added Statistical properties: are very complicated, may be assumed to be a stationary zeromean white noise sequence with uniform probability density over the range −Q2,Q2. Moreover, e(n) is assumed to be uncorrelated with the signal x(n) Variance: Autocorrelation: Crosscorrelation: Statistical properties quantization noise xQ(n) x(n) e(n) 12 ( ) 2 2 2 Q E e n e Ree(k) Ee(n k)e(n)e2 (k) Rex(k) Ee(n k)x(n) 0 Purposes: Oversampling was mentioned earlier as a technique to alleviate the need for high quality prefilters and postfilters. It can also be used to trade off bits for samples: if we sample at a higher rate, we can use a coarser quantizer Power spectral density: quantized signal xQ(n) as a noisy version of the original unquantized signal x(n) to which a noise component e(n) has been added Analysis: Consider two cases, one with sampling rate fs and B bits per sample, and the other with higher sampling rate f’s and B’ bits per sample. To maintain the same quality in the two cases, we require that the power spectral densities remain the same Oversampling e s See f f 2 ( ) 2 2 s fs for f f e s e s f f 2 2 CuuDuongThanCong.com https:fb.comtailieudientucntt5 If sampling is done at the higher rate f’s , then the total power σ’e2 of the quantization noise is spread evenly over the f’s Nyquist interval. Number of bit difference: Analysis: a saving of half a bit per doubling of L. This is too small to be useful. For example, in order to reduce a 16bit quantizer for digital audio to a 1bit quantizer, that is, ΔB = 15, one would need the unreasonable oversampling ratio of L = 230. Oversampling B B B e e L 2( ) 2 2 2 2 2 B L 0.5log2 Number of bit difference: Noise shaping quantizers reshape the spectrum of the quantization white noise into a more convenient shape. Power spectral density: Analysis: A noise shaping quantizer operating at the higher rate fs can reshape the flat noise spectrum so that most of the power is squeezed out of the fs Nyquist interval and moved into the outside of that interval. Noise shaping 2 2 2 ( ) ( ) ( ) H ( f ) f S f H f S f NS e s NS ee CuuDuongThanCong.com https:fb.comtailieudientucntt6 A typical pth order noise shaping: Small f range: Assuming: a large oversampling ratio L, we will have fs