Abstract
Discrete-time signals are, in general, infinite-length sequences of numerical values that may either arise from sampling of continuous-time signals, or be generated directly by inherently-discrete-time processes. Deterministic signals have a univocal mathematical description, so that future signal values are exactly predictable. Random signals do not allow for such a description: their treatment requires statistical tools since signal evolution cannot be exactly foreseen. This chapter introduces basic concepts related to discrete-time signals, as well as mathematical operators, called discrete-time systems, that are employed to process them. The main constraints imposed on discrete-time systems, namely linearity, time invariance, stability and causality, are introduced along with the quantities used to describe a system: the impulse response, the transfer function and the frequency response. Finite-impulse-response (FIR) and infinite-impulse-response (IIR) systems are defined. Linear convolution and the linear constant-coefficient-difference equation (LCCDE) are introduced to express the input-output relation of discrete-time systems.
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Notes
- 1.
The independent variable might, of course, also be a spatial coordinate, however, here we will assume that we are only interested in time variations.
- 2.
Signal processing systems are classified along the same lines as signals. We can thus have continuous-time systems for which both the input and the output are continuous-time signals, discrete-time systems for which both the input and the output are discrete-time signals, and digital systems, for which both the input and the output are digital signals. Strictly speaking, therefore, digital signal processing deals with the transformations of signals that are discrete in both amplitude and time. In this book, the word “digital” is used loosely as a synonym of “discrete-time”, since we do not enter into details about the implications of amplitude quantization. In the same way, we will use the word “analog” as a synonym of “continuous-time”.
- 3.
We use round and square brackets to indicate intervals, with the former meaning that the corresponding edge value is not included in the interval, and the latter meaning that the corresponding edge value is included in the interval. Intervals are referred to as open in the first case and closed in the second case. For example, \([-\pi , +\pi )\) indicates a range that includes the edge value \(-\pi \) (square bracket) but excludes the edge value \(+\pi \) (round bracket); this is an open interval; \([-\pi , +\pi ]\) would be a closed interval. Whenever an edge value is \(\pm \infty \), the interval will be open in the corresponding direction.
- 4.
Note that for real signals we could simply write energy as the integral of \(x^2[n]\). However, here and in the rest of the book we will stick to the definition given above, holding for general complex-valued signals.
- 5.
This definition of linear convolution is valid for general complex sequences.
- 6.
Note that for order M, h[n] in the above equation extends over [0, M] and has \(M+1\) samples. This notation contrasts with the usual notation we adopted for finite-length sequences, according to which we would write \([0,M'-1]\), i.e., \(M'\) samples, order \(M'-1\). We will have to be careful with this inconsistency in future discussion.
- 7.
The Fourier series and transform encountered in the theory of analog and digital signals are orthogonal signal expansions (a.k.a. representations). These expansions constitute a fairly standard topic in any undergraduate program in scientific sectors. However, a synthetic introduction to signal representations and to the related topic of vector spaces is provided in the appendix to the next chapter.
References
Haykin, S., Van Veen, B.: Signals and Systems. Wiley, New York (2002)
Mitra, S.K.: Digital Signal Processing. McGraw-Hill, Blacklick (2011)
Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice Hall, Englewood Cliffs (2009)
Proakis, J.G., Manolakis, D.G.: Digital Signal Processing. Prentice Hall, Englewood Cliffs (2006)
Shenoi, B.A.: Introduction to Digital Signal Processing and Filter Design. Wiley, New York (2007)
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Alessio, S.M. (2016). Discrete-Time Signals and Systems. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_2
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DOI: https://doi.org/10.1007/978-3-319-25468-5_2
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