![]() ![]() Many of these conversions were mentioned in chapters dealing with individual forms of description. It is obvious that the same applies to the phase frequency response, with the only difference that unlike the magnitude frequency response, the phase frequency response is an odd function, due to the way of its computation. In the period, values of the frequency transfer function are symmetric about the interval centre, i.e. If the argument of the complex exponential function is equal to ΩT s = Ω/f s, then repeating values of this function with an angular period 2π mean that they also repeat with the angular sampling frequency Ω s. This fact implies that values of the transfer function repeat periodically, with the argument value equal to integer multiples of 2π. For this reason, values of the frequency transfer function are displayed above the unit circle in the complex plane. ![]() The dependence of magnitude of values of the frequency transfer function on frequency is called the magnitude frequency response of a discrete system, whereas the dependence of argument of values of the frequency transfer function on frequency is called the phase frequency response.Īs we have already mentioned before, the substitution function for the z variable (eq.(6.5), r = 1) is a complex exponential function with a unit modulus, the values of which, in a complex plane, are displayed on a unit circle. Its argument defines the phase (or time) shift between harmonic components of input and output. The magnitude of the frequency transfer function determines the ratio between the amplitudes of harmonic components for a given frequency, which input and output sequences are composed of. Because this is again a way of external (input/output) description of linear system characteristics, the function expresses the relation between the input and output sequences of the system or, alternatively, between the harmonic sequences which both sequences are composed of. ![]() This function is called the frequency transfer function. Although it has much in common (as we will see later) with the discrete-time Fourier transform (DTFT), we do not use it to decompose sequences to simpler ones (although in principle, such interpretation would also be possible) but mainly to describe linear time-invariant systems. The Z-transform is a useful mathematical tool for the transformation of sequences into a complex plane. For this purpose, we need to get acquainted with the definition and characteristics of the Z-transform. The following text will deal with these forms of description and with what they reveal from the system characteristics. However, linear systems make various forms of description possible while maintaining a broad scope of use. The above-mentioned character of system parameters implies that linear systems represent a significant simplification – their parameters must be constant. On the other hand, the question arises whether there is another possibility of system description that would be able to reveal other interesting or useful characteristics of the studied system. the possibility to determine the values representing the behaviour of the modelled subject. The difference equation and its solution is the most important feature expected from the construction of a mathematical model, i.e. An alternative notation of the difference equation has the form of expression for the computation of the kth output sample, which employs the value of the kth sample of input sequence and previous samples of both input and output sequences up to the delay m or n, respectively. The value n determines the maximum delay for samples of the output sequence and the system order at the same time, whereas m determines the maximum delay for samples of the input sequence, which is involved in the calculation. If the parameters are constant, the system meets the superposition principle and the system is linear. In general, parameters ai and bi can be functions of both input and output variables (non-linear systems) and of time, too (time-variant systems). without an input), the difference equation is homogeneous, with the right-hand side equal to zero. U\).Where a i and b i are system parameters, x(k) are values of the input sequence and y(k) are values of the output sequence. ![]()
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