Control Systems/Transfer Functions - Wikibooks, open books for an open world
The impulse that is referred to in the term impulse response is . functions at varying frequencies (there is an analogous relationship for. A Transfer Function is the ratio of the output of a system to the input of a system, Readers who have read the Circuit Theory book will recognize the transfer function as We define the impulse response as being the relationship between the. h is called the impulse response of the system can manipulate block diagrams with transfer functions as if they were Example: feedback connection u. G consider a circuit with linear elements, zero initial conditions for inductors.
To determine an output directly in the time domain requires the convolution of the input with the impulse response. When the transfer function and the Laplace transform of the input are known, this convolution may be more complicated than the alternative of multiplying two functions in the frequency domain.
The impulse response, considered as a Green's functioncan be thought of as an "influence function": Practical applications[ edit ] In practical systems, it is not possible to produce a perfect impulse to serve as input for testing; therefore, a brief pulse is sometimes used as an approximation of an impulse. Provided that the pulse is short enough compared to the impulse response, the result will be close to the true, theoretical, impulse response.
In many systems, however, driving with a very short strong pulse may drive the system into a nonlinear regime, so instead the system is driven with a pseudo-random sequence, and the impulse response is computed from the input and output signals.
Impulse response - Wikipedia
Loudspeakers suffer from phase inaccuracy, a defect unlike other measured properties such as frequency response. The need to limit input amplitude to maintain the linearity of the system led to the use of inputs such as pseudo-random maximum length sequencesand to the use of computer processing to derive the impulse response. An interesting example would be broadband internet connections. Control systems[ edit ] In control theory the impulse response is the response of a system to a Dirac delta input.Problem No.1 on Impulse Response in Discrete Time signal Time Processing
This proves useful in the analysis of dynamic systems ; the Laplace transform of the delta function is 1, so the impulse response is equivalent to the inverse Laplace transform of the system's transfer function. Acoustic and audio applications[ edit ] In acoustic and audio applications, impulse responses enable the acoustic characteristics of a location, such as a concert hall, to be captured.
Various packages are available containing impulse responses from specific locations, ranging from small rooms to large concert halls. These impulse responses can then be utilized in convolution reverb applications to enable the acoustic characteristics of a particular location to be applied to target audio. Then, the output would be equal to the sum of copies of the impulse response, scaled and time-shifted in the same way.
The above equation is the convolution theorem for discrete-time LTI systems. For continuous-time systems, the above straightforward decomposition isn't possible in a strict mathematical sense the Dirac delta has zero width and infinite heightbut at an engineering level, it's an approximate, intuitive way of looking at the problem.
A similar convolution theorem holds for these systems: One method that relies only upon the aforementioned LTI system properties is shown here. For both discrete- and continuous-time systems, the impulse response is useful because it allows us to calculate the output of these systems for any input signal; the output is simply the input signal convolved with the impulse response function.
An LTI system's frequency response provides a similar function: Recall the definition of the Fourier transform: These scaling factors are, in general, complex numbers.
Here's where it gets better: The idea is, similar to eigenvectors in linear algebra, if you put an exponential function into an LTI system, you get the same exponential function out, scaled by a generally complex value.
This has the effect of changing the amplitude and phase of the exponential function that you put in.
This is immensely useful when combined with the Fourier-transform-based decomposition discussed above. These effects on the exponentials' amplitudes and phases, as a function of frequency, is the system's frequency response. An LTI system's impulse response and frequency response are intimately related.
- Control Systems/Transfer Functions
- The Unit Impulse Response
The frequency response is simply the Fourier transform of the system's impulse response to see why this relation holds, see the answers to this other question.