Infinite-impulse—response filter

Bialkowski, S. E., Real-Time Digital Filters Infinite Impulse Response Filters, Anal. Chem. 60, 1988, 403A- 13A. 

HR Filter (See Infinite Impulse Response Filter) IRCAM 4B, 213 4c, 214 4X, 215 ISPW, 232 IRIS... 

Infinite impulse response filter -> recursive filter... 

Many DSP concepts can be demonstrated by examples which involve a great deal of computation. A list of some of the concepts is as follows convolution, filtering, quantization effects, etc. The curriculum begins with discrete Fourier transform (DFT). DFT is derived from discrete-time Fourier transform expression. The continuous and discrete Fourier transform are covered in Signals and Systans. The flow of the topics is as follows DFT, properties of DFT, Fast Fourier Transform, Infinite Impulse Response filter and Finite Impulse Response fillers and filter structures. If the topics are linked to a project with each block of the project demonstrating the various topics of the curriculum, it is easier for the student to comprehend what is being taught. 

Infinite impulse response filter recursive filter Infrared spectroscopy spectroscopy... 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

The reverberation algorithm can be based on efficient infinite impulse response (HR) filters. 

Time Domain Analysis. Perhaps the simplest and most traditional use of a DSP is filtering. DSPs are designed to implement both Finite Impulse Response (FIR) and Infinite Impulse Response (HR) filters as fast as possible by implementing (a) a single cycle multiply accumulate instruction (b) circular addressing for filter coefficients. These two requirements can be found in all modem DSP architectures. 

It s easy to show that the impulse response of this filter for g= 1 isr" = 1.0, r, r r, r, etc. This type of response is called an exponential decay. It s easy to see why filters of this form are called Infinite Impulse Response (HR) filters, because for a nonzero r, the output technically never goes exactly to zero. If r is negative, the filter will oscillate positive and negative each sample, corresponding to even and odd powers of r. This is called an exponential oscillation. If the magnitude of r is greater than one, the filter output will grow without bound. This condition is called instability, and such filters are called unstable. 

The infinite impulse response or HR filter has the difference equation ... 

A digital filter with impulse response having infinite length is called an infinite impulse response (HR) filter. An important class of HR filters can be described by the difference equation... 

Consider the impulse response of the second order HR filter a = —1.8,02 = 1, shown in figure 10.15a. Just as in the first order case, this filter has an infinite response, but this time takes the form of a sinusoid. This shows the power of the HR filter - with only a few terms it can produce a quite complicated response. The filter a = —1.78,02 = 0.9, has its impulse response shown in Figure 10.15b. We can see that it is a decaying sinusoid we have in effect combined the characteristics of examples 10.14b and 10.15a. The key point to note is that in all cases, the input is the same, ie a unit impulse. The characteristics of the output are governed solely by the filter -after all the input has no periodicity or decay factor built in. If we choose a slightly different set... 

Figure 10.20 Plots of first order HR filter, with a=0,8, 0,7, 0,6 and 0.4. As the length of decay increases, the frequency response becomes sharper. Because only a single coefficient is used, there will be one pole, which will always lie on the real-axis. As ai 1, the impulse response will have with no decay and the pole will lie 1.0. Because the pole lies on the unit circle, it will lie in the frequency response, and hence there will be an infinite value for frequency at this point in the spectmm.

Some considerations that are necessary in the design of the filter include whether the filter is finite duration impulse response or infinite duration impulse response (HR). Also the choice of optimality... 

If we give this filter the unit impulse as an input, then we can calculate the output for any n after 0. The output is shown in Figure 10.14b. First of all we observe that the output is not of a fixed duration in fact it is infinite as the name of the filter implies. Secondly we observe that the response takes the form of a decaying exponential. The rate of decay is governed by the filter coefficient, a. (Note that if > 1 the output will grow exponentially this is general undesirable and such filters are called unstable.)... 

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