5. Spectral Analysis of Deterministic Discrete-Time Signals

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.