CH2: SIGNAL SAMPLING AND QUANTIZATION

Digital Signal Processing

Fundamentals and Applications 2nd Edition

Li Tan & Jean Jiang

SAMPLING OF CONTINUOUS SIGNAL

ADC unit samples the analog signal, quantizes the sampled signal, and encodes thequantized signal level to the digital signal.

Infinite points cannot be processed by the digital signal processor, since they require aninfinite amount of memory and infinite number of processing power for computation.

SAMPLING can solve the problem by taking samples at a fixed time interval.

SAMPLE AND HOLD each sample maintains its voltage level during the samplinginterval (T) to give the ADC enough time to convert it.

Aliasing unwanted signals in the desired frequency band

Analog signal can be in theory perfectly recovered as long as the sampling rate is at least twice as large as the highest-frequency component of the analog signal to be sampled. (fs > = 2fmax)

Shannon Sampling Theorem: For a uniformly sampled DSP system, an analog signal can be perfectly recovered as long as the sampling rate is at least twice as large as the highest-frequency component of the analog signal to be sampled.

SUMMARY

SIGNAL RECONSTRUCTION

If an analog signal with a frequency (f) is undersampled, the aliasing frequency component faliasin the baseband is simply given by the following expression: falias = fs f

Anti-Aliasing Filter

By applying this, we limit the input analog signal, so that all the frequency components are less than the folding frequency (half of the sampling rate).

To control the aliasing noise: Use a higher order lowpass filter

Increase the sampling rate

For illustrative purpose, Butterworth filter will be used. It can be extended to other types of filter such as the Chebyshev filter.

Butterworth magnitude frequency responsewith an order of n is given by:

Sample and Hold Effect help us design the anti-image filter

To overcome the sample and hold effect:

1. Using an equalizer, the sample-and-hold shaping effect can be compensated, whosemagnitude response is opposite to the shape of the hold circuit magnitude frequencyresponse.

2. Increase the sampling rate by oversampling and interpolation method when a highersampling rate is available at the DAC. Using the interpolation will increase the samplingrate without affecting the signal bandwidth, so that the baseband spectrum and its imagesare separated further apart and a lower-order anti-aliasing filter can be used.

3. Change the DAC configuration and perform digital pre-equalization using a flexible digitalfilter whose magnitude frequency response is against the spectral shape effect due to thehold circuit. In this way, the spectral shape effect can be balanced before the sampledsignal passes through the hold circuit. Finally, the anti-image filter will remove the rest ofimages and recover the desired analog signal.

The specifications for anti-aliasing filter designs are similar to anti-image (reconstruction) filtes, except for their stopband edges.

Anti-aliasing filter designed to block the frequency components beyond the folding frequency before the ADC operation

Reconstruction filter designed to block the frequency components beginning at the lower edge of the first image after the DAC

ANALOG-DIGITAL CONVERSION, DIGITAL-ANALOG CONVERSION

During the ADC process, amplitudes of the analog signal to be converted have infinite precision.

Quantization continuous amplitude is converted to digital data with finite precision

Several way to implement ADC: Flash ADC

Successive approximation ADC

Sigma-delta ADC

Flash ADC offers the advantage of high conversion speed, since all bits are acquired at the same time.

Quantization Error obtained by subtracting the original analog voltage from the recovered analog voltage

Quantization Process the process of converting analog voltage with infinite precision to finite precision

Unipolar Quantizer deals with analog signals ranging 0 volt to a positive reference voltage

Bipolar Quantizer deals with analog signals ranging from a negative reference to a positive reference

Xmax & Xmin: maximum and minimum value of the analog signal x

L: number of quantization levels

m: number of bits used in ADC

triangle: step size of the quantizer or the ADC resolution

Xq: indicates the quantization level

i: index corresponding to the binary code

DAC PROCESS

DAC unit takes the binary codes from the DS processor.

It converts the binary code using the zero-order hold circuit to reproduce the sample-and-hold signal.

The recovered sample-and-hold signal is further processed using the anti-image filter.

Finally, the analog signal is produced.

When the DAC outputs the analog amplitude xq with finite precision, it introduces quantization error defined as: