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:
“The impulse to perfection cannot exist where the definition of perfection is the arbitrary decision of authority. That which is born in loneliness and from the heart cannot be defended against the judgment of a committee of sycophants. The volatile essences which make literature cannot survive the clichés of a long series of story conferences.”
—Raymond Chandler (1888–1959)
“I’ll never forget my father’s response when I told him I wanted to be a lawyer. He said, “If you do this, no man will ever want you.””
—Cassandra Dunn (b. c. 1931)
“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)
“For me being a poet is a job rather than an activity. I feel I have a function in society, neither more nor less meaningful than any other simple job. I feel it is part of my work to make poetry more accessible to people who have had their rights withdrawn from them.”
—Jeni Couzyn (b. 1942)