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no spectral distortion if N ≥ M = T x /T s . Equivalently, there will be no spectral distortion if

1
1
1
1
∆f = --- = --------- ≤ ---------- = -----
T NT s MT s T x
Parameters for the DFT processing of a sampled continuous signal must be carefully selected to avoid
spectral distortion due to aliasing or data truncation. Assuming a window width of T x seconds and that
the signal has a maximum bandwidth of B hertz, then based on the sampling theorem we would have
negligible aliasing, provided F s = 1/T s ≥ 2B. Spectral leakage due to sharp data truncation is avoided
provided the frequency resolution is selected to satisfy 1/∆f = T ≥ T x . Consequently, spectral distortion
due to aliasing and spectral leakage can be avoided if the length of the DFT is selected to satisfy N =
F s /∆f ≥ 2BT x .

References
1. Soliman, S.S., and Srinath, M.D., Continuous and Discrete Signals and Systems, Prentice-Hall, 1998.
2. O’Flynn, M., and Moriarty, E., Linear Systems: Time Domain and Transform Analysis, Wiley, 1987.
3. Lathi, B.P., Modern Digital and Analog Communication Systems, Oxford University Press, Third
Edition, 1998.
4. Proakis, J.G., and Salehi, M., Communication Systems Engineering, Prentice-Hall, 1994.
5. Taylor, F.J., Principles of Signals and Systems, McGraw-Hill, 1994.
6. Carlson, G.E., Signal and Linear System Analysis, Wiley, Second Edition, 1998.
7. Proakis, J.G., and Manolakis, D.G., Digital Signal Processing: Principles, Algorithms, and Applications,
Prentice-Hall, 1996.
8. Oppenheim, A.V., and Schafer, R.W., with Buck, J.R., Discrete-Time Signal Processing, Prentice-Hall,
Second Edition, 1999.
9. Houts, R.C., Signal Analysis in Linear Systems, Saunders College Publishing, 1991.
10. Ziemer, R.E., Tranter, W.H., and Fannin, D.R., Signals and Systems: Continuous and Discrete, MacMillan
Publishing Company, Third Edition, 1993.
11. Orfanidis, S.J., Introduction to Signal Processing, Prentice-Hall, 1996.
12. Haykin, S., and Veen, B.V., Signals and Systems, Wiley, 1999.
13. Taylor, F., and Mellot, J., Hands-On Digital Signal Processing, McGraw-Hill, 1998.

23.2 z Transform and Digital Systems

Rolf Johansson
A digital system (or discrete-time system or sampled-data system) is a device such as a digital controller
or a digital filter or, more generally, a system intended for digital computer implementation and usually
with some periodic interaction with the environment and with a supporting methodology for analysis
and design. Of particular importance for modeling and analysis are recurrent algorithms—for example,
difference equations in input–output data—and the z transform is important for the solution of such
problems.
The z transform is being used in the analysis of linear time-invariant systems and discrete time
signals—for example, for digital control or filtering—and may be compared to the Laplace transform as
used in the analysis of continuous-time signals and systems, a useful property being that the convolution
of two time-domain signals is equivalent to multiplication of their corresponding z transforms. The z
transform is important as a means to characterize a linear time-invariant system in terms of its pole–zero
locations, its transfer function and Bode diagram, and its response to a large variety of signals. In addition,
it provides important relationships between temporal and spectral properties of signals. The z transform
generally appears in the analysis of difference equations as used in many branches of engineering and
applied mathematics.

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