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Signal Processing revision notes
CD player
CD read head reads data at constant speed (disc speed varies)
Pits and lands: pit is wavelength deep. Area of spot arranged so that area striking pit = area striking surrounding land. Beam cancels out when over pit. Transition between pit and land = binary 1.
CD audio is 16-bit, 48kHz.
CD-R uses pre-groove to guide cutting head. Reflectivity of photosensitive dye is changed by laser heating.
CD-RW: dye used is crystalline and has variable reflectivity (phase-change).
Oversampling principle:
Upsample the signal, typically by a factor of 4 or more. Digitally filter to remove the images of the original lower sampling rate. Use lower-order analogue filter to reconstruct signal- avoids need for complex high-order analogue reconstruction filter.
Philips digital filter is tuned to compensate for analogue reconstruction filter, too (analogue is 3rd order Bessel). Irregularities in stopband caused by filter coefft rounding.
Efficient interpolation obtained using delay blocks of L samples (L is oversampling factor) and using L sets of coefficients on each multiplier.
Noise shaping- getting 16-bit performance from 14-bit DAC.
Noise shaping moves the DACs quantisation noise up the spectrum so that it is removed by the reconstruction filter. It can be taken to extremes: 1-bit DACs are used in modern CD players as they dont suffer from differential nonlinearity (uneven quantisation levels). However, the jitter performance degrades.
Noise shaping (Philips DAC):
upsample by 4 (28-bit output at 176.4kHz)
feed the 14 MSBs into 14-bit DAC.
feed the 14 LSBs back and add them to next sample coming out of interpolator.
Aperture effect:
To find the frequency response of a DAC, take the FT of its impulse response.
Impulse response of a DAC is a square pulse of width ( where:
EMBED Equation.3
where a is the percentage aperture and T is the sampling interval.
Since the impulse response is a square pulse, the frequency response is a sinc function. As a reduces, the main lobe becomes lower and wider and in the limit, a DAC with an aperture of 0 produces a flat frequency response.
Sample rate conversion:
For an integer ratio, interpolate and then decimate. Note that you must filter before decimation to ensure that aliases are removed. For a sample-rate-converter, the output filter of the interpolator and the input filter of the decimator may be combined into a single filter.
Dealing with incommensurate clocks
Use an interpolator attached to system A to drive a latch clocked by system B. Then decimate and feed the output into system B. This ensures that the duplicated samples will be much less noticeable, although they occur more often.
Continuously variable sample rate conversion.
Used for replay from variable-speed tape, or for dealing with small changes in sampling frequency. Takes advantage of the fact that each individual sample generates a sinc function. Continuous description of a band-limited signal can be found by summing these samples over time. We only consider the R nearest functions on each side.
EMBED Equation.3
Timing Jitter
Jitter noise power is:
EMBED Equation.3
On the whole, quantisation noise should equal jitter noise.
The fast Fourier transform (FFT)
The DFT requires N2 complex multiplications. For the FFT, this is N log2N. Input data must be pre-shuffled and then butterfly equations can be applied.
Noise in Digital Systems
Using 2s complement, the maximum and minimum values are adjacent. So if there is a numeric over or underflow the resulting distortion is significant. DSP chips usually implement saturation arithmetic. You can check for overflows by comparing sign bits, or (better) by using carry in/carry out bits and an XOR gate.
Most noise in digital filters is caused by rounding errors. Multiplications in filters increase the wordlength, but at some point this must be rounded off. The rounding only usually affects the recursive part of the filter and is coloured by it. This can be avoided by applying a noise shaping technique, using an extra set of taps and coefficients to feed the error signal back into the adder. This then flattens the noise spectrum at the expense of needing twice as much hardware.
Dithering
For small signals, quantisation noise can become very significant and introduce harmonic distortion. Dither is adding noise (usually with an rms voltage of 1/3 quantisation level) to the signal before quantising. This reduces harmonic distortion at the expense of adding more white noise to the signal. This is more acceptable to the listener.
Coefficient quantisation
Limiting the wordlength of the coefficients changes their values and thus the frequency response of the filter. In extreme cases this could destabilise the filter by moving the poles outside the unit circle. Changing the filter structure (using a coupled-form structure) can radically improve this by moving the way that poles are distributed without altering the frequency response.
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