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william-chapman
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Digital to Analogue Conversion
• Natural signals tend to be analogue
• Need to convert to digital
)2sin()( fttx
)2sin()( sfntnx
Sampling• Need to hold the signal steady for long enough
to enable the A to D converter to generate an output
SampleandHold
SampleandHold
AnalogueTo digitalconverter
AnalogueTo digitalconverter
• Quantization converts continuous value to a discrete (usually integer) value
• Output value is rounded so accuracy lost
• Maximum quantization error of ±0.5 lsb
• Error is combined with signal as noise
Quantization accuracy
V30.5 2
volts2
2
range voltagefull valuelsb
or
7.81mV 2
volts2
2
range voltagefull valuelsb
:convertersbit 16 and 8for volt then1 is rangeinput If
16bits num
8bits num
• Least significant bit determines accuracy. So for a 2 Volt peak to peak signal, an 8 bit converter can accurately represent multiples of 7.81mV but anything in between will be rounded
Quantization error
Error is at most ±1/2 an lsb, or ±3.905 mV for the 8 bit converter or ±15.25µV in the 16 bit case
Quantization error
• Relatively small signal changes are subject to severe quantization errors
• Quantization error creates steps
• Steps create distortion which is visible in the frequency domain
• Noise shown on dB scale as it is relatively small compared to the signal
Sampling theory
• Signal needs to be sampled at twice the speed of the fastest change to be captured
• Shannon or the Nyquist sampling theorem, (authors of 1940s papers)
• Theorem states that a continuous signal can be properly sampled, only if it does not contain frequency components above one-half of the sampling rate
Correctly sampled signal• Signal frequency is 0.09 of the sample rate
(i.e. sample rate is about 11x signal freq)
• e.g. 90Hz signal sampled at 1kHz
Sample rate still ok• Signal frequency is 0.31 of the sample rate
(i.e. sample rate about 3x signal freq)
• 3.2 samples / cycle but freq still preserved
Improper Sampling• Signal frequency is 0.95 of the sample rate (i.e.
sample rate only slightly higher than signal freq)
• Only 1.05 samples per cycle. Produces a 0.05Hz alias signal which is mixed with the original
Sidebands
• Sampling a signal is effectively multiplication of signals in the time domain
• Multiples of the sample frequency are produced as well as sum and difference frequencies (sidebands)
No sampling no sidebands• Time domain to frequency domain of an
analogue signal
Sampled signal produces sidebands
Incorrectly sampled signal• Breaching Nyquist causes aliasing with
overlapping sidebands
Simulated sampling
• Using a sample rate of 1kHz, the frequency spectrum with noise was calculated from:
)3002sin()1202sin()502sin()( sss ntntntnf
• Then modified to illustrate aliasing by changing 300Hz signal to 800Hz:
)8002sin()1202sin()502sin()( sss ntntntnf
)3002sin()1202sin()502sin()( sss ntntntnf
Correct sample rate
Aliasing at 200Hz
)8002sin()1202sin()502sin()( sss ntntntnf
Anti-alias filter• Frequencies higher than those of interest (such
as noise) need to be blocked before sampling. Use an analogue low pass filter
DSP system
AnalogueAnti-alias
Filter
AnalogueAnti-alias
Filter
Analogueto digitalconverter
Analogueto digitalconverter
SampleandHold
SampleandHold DSPDSP
Digitalto analogueconverter
Digitalto analogueconverter
AnalogueReconstruction
Filter
AnalogueReconstruction
Filter
• Low pass input filter removes F > 0.5F(S)
• Reconstruction filter removes high frequency F(S) multiples
Sound Blaster block diagram