By J. M Ziman
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Extra info for Electrons in metals;: A short guide to the Fermi surface
15) where f (k)(0) is the kth derivative of f(t) at t = 0. The initial conditions for f(t) are necessary to its Laplace transform just as they are necessary for the complete solution of an ordinary differential equation. For the digital case, we must first employ a formula for carrying forward initial conditions in the z-transform of a time-advanced signal. 16 can be easily proved from the definition of the z-transform. 17) where T is the sample increment. 16 gives the z-transform of the first derivative.
This makes generating large sample words at lower frequencies very accurate. However, the noise-shaping effect also makes the upper end of the signal frequency response roll off well below the Nyquist rate. DSC manufacturers correct for this using a digital filter to restore the high-frequency response, but this also brings a time delay to the DSC output sample. This will be discussed in more detail in Chapter 3 in the sections on digital filtering with finite impulse response (FIR) filters. 8 A delta modulator is used to convert an analog voltage to a 1-bit “bitstream” signal where the amplitude of the signal is proportional to the number of 0s and 1s in a given section of the bitstream.
1 gives a partial listing of some useful Laplace transforms and the corresponding z-transforms assuming regularly sampled data every T seconds (fs = 1/T samples/s). 1 are how some of the z-transform magnitudes scale with the sample interval T. It can be seen that the result of the scaling is that the sampled impulse responses may not match the inverse z-transform if a simple direct s-to-z mapping is used. Since adaptive digital signal processing can be used to measure and model physical system responses, we must be diligent to eliminate digital system responses where the amplitude depends on the sample rate.
Electrons in metals;: A short guide to the Fermi surface by J. M Ziman