Impulse Response and Transfer Function
The impulse response h of a linear time-invariant causal filter specifies the output that the filter would produce if it were to receive an input consisting of a single impulse at time 0. An "impulse" in a continuous time filter means a Dirac delta function; in a discrete time filter the Kronecker delta function would apply. The impulse response completely characterizes the response of any such filter, inasmuch as any possible input signal can be expressed as a (possibly infinite) combination of weighted delta functions. Multiplying the impulse response shifted in time according to the arrival of each of these delta functions by the amplitude of each delta function, and summing these responses together (according to the superposition principle, applicable to all linear systems) yields the output waveform.
Mathematically this is described as the convolution of a time-varying input signal x(t) with the filter's impulse response h, defined as:
The first form is the continuous-time form which describes mechanical and analog electronic systems, for instance. The second equation is a discrete-time version used, for example, by digital filters implemented in software, so-called digital signal processing. The impulse response h completely characterizes any linear time-invariant (or shift-invariant in the discrete-time case) filter. The input x is said to be "convolved" with the impulse response h having a (possibly infinite) duration of time T (or of N sampling periods).
The filter response can also be completely characterized in the frequency domain by its transfer function, which is the Fourier transform of the impulse response h. Typical filter design goals are to realize a particular frequency response, that is, the magnitude of the transfer function ; the importance of the phase of the transfer function varies according to the application, inasmuch as the shape of a waveform can be distorted to a greater or lesser extent in the process of achieving a desired (amplitude) response in the frequency domain.
Filter design consists of finding a possible transfer function that can be implemented within certain practical constraints dictated by the technology or desired complexity of the system, followed by a practical design that realizes that transfer function using the chosen technology. The complexity of a filter may be specified according to the order of the filter, which is specified differently depending on whether one is dealing with an IIR or FIR filter. We will now look at these two cases.
Read more about this topic: Linear Filter
Famous quotes containing the words impulse, response, transfer and/or function:
“If family violence teaches children that might makes right at home, how will we hope to cure the futile impulse to solve worldly conflicts with force?”
—Letty Cottin Pogrebin (20th century)
“From time to time I listen to what you are saying, just in case a response is needed.”
—Mason Cooley (b. 1927)
“If it had not been for storytelling, the black family would not have survived. It was the responsibility of the Uncle Remus types to transfer philosophies, attitudes, values, and advice, by way of storytelling using creatures in the woods as symbols.”
—Jackie Torrence (b. 1944)
“Advocating the mere tolerance of difference between women is the grossest reformism. It is a total denial of the creative function of difference in our lives. Difference must be not merely tolerated, but seen as a fund of necessary polarities between which our creativity can spark like a dialectic.”
—Audre Lorde (1934–1992)