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Explain the Impact of the Quantization Process.
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Solution:

Analog input to a digital signal processor is converted to a fixed word-length digital signal using an ADC. In Section 7.3, we saw the various word lengths for the I/O signals for various TI DSP processors.

We have also seen that the coefficients are stored in registers with fixed register lengths.

Thus the I/O signals are not processed and/or stored in the processor with infinite precision as this would require infinitely large memory.

Both the signals and the coefficients are quantized into fixed word lengths in order to fit into the available storage.

During a multiplication operation, a b-bit signal sample is multiplied with a b-bit coefficient to produce a 2b-bit product. Such a product may be stored or sent to a DAC converter after the 2 b-bit words are shortened or truncated to b bits.

The impact of quantization of the input signal, the coefficients, and the result of intermediate operations are that the signal at the output will be different from what is expected theoretically.

This is because quantization introduces errors in the input signal, in the coefficients, and in the products of the intermediate operations.

The error that is introduced to the output signal depends on how the numbers and the arithmetic operations are done.

The errors will be different depending on whether fixed-point or floating-point number representation and arithmetic operations are used.

The errors will also be influenced by the way the negative numbers are represented in fixed-point representation.

Figure 7.3 shows a signal x(t) that is quantized through a truncation to obtain the signal Q(x(t)). The quantization error is given by,

$\varepsilon_t=Q(x)-x\\$

In the section below we show the range of the truncation errors for the different types of arithmetic representation.

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