Abstract
In this chapter we will describe the spectral representation of a discrete-time deterministic signal. Even if intrinsically the signal has infinite length, we can only observe it through a window of finite width, and we cannot compute its DTFT, provided it exists, but only the DFT of the segment framed by the window. The consequences of these limitations are investigated through examples using sinusoidal signals, and the issues of leakage and loss of resolution are examined. A description of the main windows used in spectral analysis follows. We then move to more conceptual topics. Deterministic bounded signals can be energy signals or power signals. For energy signals we can use the squared magnitude of the DTFT to define the energy spectrum, describing how the energy of the signal distributes over frequency. The energy spectrum can equivalently be defined as the DTFT of the autocovariance (AC) sequence, quantifying the signal’s self-similarity. On the other hand, the spectral representation of power signals, i.e., signals with infinite energy but finite average power, necessarily passes through the transform of the AC sequence of the power signal, provided that the AC has finite energy.
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- 1.
\(20\log _{10}a=10\log _{10}a^2\) is the square of a expressed in dB.
- 2.
Often, the Von Hann window is called hanning, with the initial letter in lower case.
- 3.
The name of this theorem refers to Norbert Wiener (1894–1964) and Aleksandr Khinchin (1894–1959). Norbert Wiener proved this theorem for the case of a deterministic function in 1930 (Wiener 1930); Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published it in 1934 (Khinchin 1934). Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914 (see Jerison et al. 1997). Note that the name of the second author of the theorem, a Russian mathematician, is sometimes transliterated from the Cyrillic alphabet as “Khintchine”.
- 4.
Note that the reality of \(P_{xx}(\mathrm{{e}}^{\mathrm {j}\omega })\) suggests the possibility of denoting PSD simply by \(P_{xx}(\omega )\), rather than by \(P_{xx}(\mathrm{{e}}^{\mathrm {j}\omega })\).
- 5.
For complex sequences, \(r_{xy}[l]\) is the linear convolution of the first sequence with the folded- and complex-conjugated version of the second sequence. We consider real signals in order to simplify the notation; if we considered complex signals, all the formulas for correlation should contain a conjugation sign, e.g., \(r_{xy}[l]=\sum _{n=-\infty }^{+\infty } x[n] y^*[n-l]\). This is necessary, for instance, if we are dealing with a complex exponential signal.
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Alessio, S.M. (2016). Spectral Analysis of Deterministic Discrete-Time Signals. 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_5
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DOI: https://doi.org/10.1007/978-3-319-25468-5_5
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