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ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals

This topic covers the methods of sampling and reconstruction in digital signal processing, including concepts like quantization, aliasing, and the Nyquist limit. Learn about the relationship between continuous-time and discrete-time signals, and the importance of anti-alias filters. Gain insights into the Spectra of Sampled Signals and the Nyquist Frequency theorem, as well as the impact of quantization on signal quality. Explore the challenges of coefficient quantization and its effects on filter constants.

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ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals

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  1. ELEN E4810: Digital Signal ProcessingTopic 11: Continuous Signals • Sampling and Reconstruction • Quantization Dan Ellis

  2. WORLD 1. Sampling & Reconstruction • DSP must interact with an analog world: A to D D to A x(t) x[n] y[n] y(t) DSP ADC DAC Anti- alias filter Sample and hold Reconstruction filter Sensor Actuator Dan Ellis

  3. Sampling: Frequency Domain • Sampling: CT signalDT signal by recording values at ‘sampling instants’: • What is the relationship of the spectra? • i.e. relateand Discrete Continuous Sampling period T  samp.freq. WT = 2p/T rad/sec W in rad/second CTFT DTFT w in rad/sample Dan Ellis

  4. t Sampling • DT signals have same ‘content’ as CT signals gated by an impulse train: • gp(t) = ga(t)·p(t) is a CT signal with the same information as DT sequence g[n] gp(t) ga(t) t t ‘sampled’ signal:still continuous - but discrete values CT delta fn p(t) T 3T Dan Ellis

  5. Spectra of sampled signals • Given CT • Spectrum • Compare to DTFT • i.e. by linearity W p/T= WT/2 w p Dan Ellis

  6. Spectra of sampled signals • Also, note thatis periodic, thus has Fourier Series: • But so shift in frequency - scaled sum of shifted replicas of Ga(jW)by multiples of sampling frequency WT Dan Ellis

  7. CT and DT Spectra So: or DTFT CTFT ga(t) is bandlimited @ WM M W -WM WM  W -2WT -WT WT 2WT = shifted/scaled copies M/T W  w -4p -2p 2p 4p Dan Ellis

  8. Aliasing • Sampled analog signal has spectrum: • ga(t) is bandlimited to ± WM rad/sec • When sampling frequency WT is large...  no overlap between aliases  can recover ga(t) from gp(t) by low-pass filtering Ga(jW) “alias” of “baseband” signal Gp(jW) W -WM WT -WT WM WT -WM -WT +WM Dan Ellis

  9. The Nyquist Limit • If bandlimit WM is too large, or sampling rate WT is too small, aliases will overlap: • Spectral effect cannot be filtered outcannot recoverga(t) • Avoid by: • i.e. bandlimit ga(t) at ≤ Gp(jW) W WM WT -WT -WM WT -WM Sampling theorum Nyquistfrequency Dan Ellis

  10. Anti-Alias Filter • To understand speech, need ~ 3.4 kHz  8 kHz sampling rate (i.e. up to 4 kHz) • Limit of hearing ~20 kHz  44.1 kHz sampling rate for CDs • Must remove energy above Nyquist with LPF before sampling: “Anti-alias” filter ‘space’ forfilter rolloff A to D ADC Anti-alias filter Sample & hold M Dan Ellis

  11. Sampling Bandpass Signals • Signal is not always in ‘baseband’ around W = 0 ... may be some higher W: • If aliases from sampling don’t overlap, no aliasing distortion, can still recover • Basic limit: WT/2 ≥bandwidth DW M W WL WH -WH -WL Bandwidth DW = WH - WL M/T W 2WT -WT WT/2 WT Dan Ellis

  12. Reconstruction • make a continuous impulse train gp(t) • lowpass filter to extract basebandga(t) Ga(ejw) • To turn g[n]back to ga(t): • Ideal reconstruction filter is brickwall • i.e. sinc - not realizable (especially analog!) • use something with finite transition band... g[n] n ^ w p ^ Gp(jW) ^ gp(t) ^ t W WT/2 ^ Ga(jW) ^ ga(t) ^ t W Dan Ellis

  13. 2. Quantization • Course so far has been about discrete-time i.e. quantization of time • Computer representation of signals also quantizes level (e.g. 16 bit integer word) • Level quantization introduces an error between ideal & actual signal noise • Resolution (# bits) affects data size  quantization critical for compression • smallest data  coarsest quantization Dan Ellis

  14. Quantization • Quantization is performed in A-to-D: • Quantization has simple transfer curve: Quantized signal Quantization error Dan Ellis

  15. + Quantization noise • Can usually model quantization as additive white noise: i.e. uncorrelated with self or signal x ^ x[n] x[n] - e[n] bits ‘cut off’ by quantization; hard amplitude limit Dan Ellis

  16. Quantization SNR • Common measure of noise is Signal-to-Noise ratio (SNR) in dB: • When |x| >> 1 LSB, quantization noise has ~ uniform distribution: signal power noise power (quantizer step = e) Dan Ellis

  17. Quantization SNR • Now, sx2 is limited by dynamic range of converter (to avoid clipping) • e.g. b+1 bit resolution (including sign)output levels vary -2b·e .. (2b-1)e where full-scale range depends on signal i.e. ~ 6 dB SNR per bit Dan Ellis

  18. Coefficient Quantization • Quantization affects not just signalbut filter constants too • .. depending on implementation • .. may have different resolution • Some coefficients are very sensitive to small changes • e.g. poles near unit circle high-Q pole becomes unstable Dan Ellis

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