Advanced Signals and Systems – Part 2: Discrete Signals and Random Processes Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical Engineering and Information Engineering Digital Signal Processing and System Theory Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes
Contents of the Lecture Entire Semester: Introduction Discrete signals and random processes Spectra Discrete systems Idealized linear, shift-invariant systems Hilbert transform State-space description and system realizations Generalizations for signals, systems, and spectra Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-2
Contents of this Part Discrete Signals and Random Processes Definitions Description of stochastic processes Examples Simple operations on stochastic processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-3
Discrete Signals and Random Processes Definitions – Part 1 Periodic and Non-Periodic Signals: A signal is called periodic if the following conditions holds: If there is no repetition, (i.e. ), then, the signal is non-periodic. The concept is easily extended to – dimensional signals with multiple periods with . Remark: In the continuous-signal domain, the function is the prototype of a periodic signal, with period . The corresponding sequence is not a-priori periodic – see conditions contained in the equation above! Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-4
Discrete Signals and Random Processes Definitions – Part 2 Energy and power signals: For closely modeling the physically defined values of continuous signals, we define the instantaneous/local power : the instantaneous/local energy : the total signal energy : the average power : Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-5
Discrete Signals and Random Processes Definitions – Part 3 Energy and power signals (continued): For periodic signals the definition of the average power turn into This is the average in any period. Signals for which the sums in the energy definitions do exist ( ), are termed signals of finite energy or, sloppily, energy signals . Signals for which the energy definitions cannot, but the power definitions can be evaluated, are termed signals of finite power or, sloppily, power signals . Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-6
Discrete Signals and Random Processes Definitions – Part 4 Deterministic and stochastic signals: If is determined by (possibly non- uniquely) and if the relation can be “evaluated”, then is deterministic . If is is of random character or if the relation cannot be “evaluated” despite of a non-random character, then is treated as stochastic . A typical example of a stochastically treated, though actually deterministic signal is speech : Contents are fixed by thoughts, the generation is determined by the “data transmission” and the physiological apparatus – but cannot be described or evaluated by any means. Another example is the error resulting from quantization of a deterministic signal (even for a sinusoid). Assume, e.g. a linear quantizer with four bits. This means: bit quantizer, leading to quantization intervals of a constant size (linear quantizer = straight line through the "stair- case“ characteristic). Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-7
Discrete Signals and Random Processes Definitions – Part 5 Deterministic and stochastic signals (continued): The quantization error is determined as If is known, i.e. deterministic , also the quantization error is deterministic ! With limitation (might also be But: Difficult evaluation even for a different) sinusoid signal (Bessel functions, …), rather impossible evaluation for more general signals! Therefore : the quantization error is very often modeled as a random process , termed quantization noise . Within a quantization interval, all amplitudes are assumed to be (Sign-magnitude characteristic) “ equally probable ”. Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-8
Discrete Signals and Random Processes Description of Stochastic Processes – Part 1 Probability and probability-density function: Definition of the probability function where is called probability and as well as must be real! For reasons of probability definitions (see theory/theorems of statistics) we have which must be monotonically non-decreasing over . The probability density function (pdf) is defined as This is a non-negative function. Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-9
Discrete Signals and Random Processes Description of Stochastic Processes – Part 2 Relations and extensions: The relation between the probability function and the probability density function can be modified (inverted) according to In addition we get If is a multidimensional or complex variable or process, e.g. with the definitions “above” (see slides before) need to be extended. Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-10
Discrete Signals and Random Processes Description of Stochastic Processes – Part 3 Relations and extensions (continued): A bi-variate probability function is defined as If the two processes can assumed to be stationary , the probability should only depend on the difference between the two sampling points : A bi-variate probability density function is defined as Again this is a non-negative function: Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-11
Discrete Signals and Random Processes Description of Stochastic Processes – Part 4 Relations and extensions (continued): This can be easily extended to multi-variate probability (density) functions. Please have a look in the corresponding literature. We will continue with bi-variate functions. The relation between probability function and the probability density function can be inverted according to As in the one-dimensional case we get also (volume under ). Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-12
Discrete Signals and Random Processes Description of Stochastic Processes – Part 5 Relations and extensions (continued): One might ask how to connect uni-variate quantities with bi- or multi-variate functions : From 2-D quantities to 1-D descriptions: marginal probability (density) (for all or only for special , correspondingly). From 1-D quantities to 2-D descriptions: Only for special cases (will be treated soon) As a second question the dependency on is of interest. To clarify this we will introduce the concept of stationarity: If the statistical properties of a one-dimension process do not depend on the (absolute) time index, the process is called stationary . This means that the probability density function does not depend on the time index . Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Digital Signal Processing and System Theory| Advanced Signals and Systems| Discrete Signals and Random Processes Slide II-13
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