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**Design of Digital IIR Filter**

DR. Wajiha Shah

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**Digital Filter convolver.**

A digital filter is simply a discrete-time, discrete-amplitude convolver. Filtering is in essence the multiplication of the signal spectrum by the frequency domain impulse response of the filter.

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**Analog and Digital Filters**

An analog filter is inherently more size-and power-efficient, although more component-sensitive, if it can be implemented in a straightforward manner. In general, as signal frequency increases, the disparity in efficiency increases. Characteristics of applications where digital filters are more size and power efficient than analog filters are: linear phase, very high stop band attenuation, very low pass band ripple. The filter’s response must be programmable or adaptive. The filter must manipulate phase and, very low shape factors. A digital filter’s shape factor is the ratio of the filter’s pass band width plus the filter’s transition band width to the filter’s pass band width.

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**Conceptual Differences**

Frequency-Domain Versus Time Domain Thinking Thinking about analog filters, most engineers are comfortable in the time domain. For example, the operation of an RC lowpass filter can easily be envisioned as a capacitor charging and discharging through a resistor. Likewise, it is easy to envision how a negative-feedback active filter uses phase shift as a function of frequency, which is a time domain operation.

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Contd… A digital filter is better conceptualized in the frequency domain. The filter implementation simply performs a convolution of the time domain impulse response and the sampled signal. A filter is designed with a frequency domain impulse response which is as close to the desired ideal response as can be generated given the constraints of the implementation. The frequency domain impulse response is then transformed into a time domain impulse response which is converted to the coefficients of the filter.

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Filters Working Filters work by using one or both of the following methods: Delay a copy of the input signal (by x number of samples), and combine the delayed input signal with the new input signal. (Finite Impulse Response, FIR, or feedforward filter) Delay a copy of the output signal (by x number of samples), and combine it with the new input signal. (Infinite Impulse Response, IIR, feedback filter)

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IIR Filters IIR filters have one or more nonzero feedback coefficients. That is, as a result of the feedback term, if the filter has one or more poles, once the filter has been excited with an impulse there is always an output.

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**Filter’s Impulse Response**

An impulse is a very short pulse—a waveform that has significant amplitude only for a very short time. (usually unipolar) For filters, we use a one-sample pulse, or unit impulse. The response of the filter to the unit impulse is the filter’s Impulse Response (IR).

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Types of Filters Types: Lowpass Highpass Bandpass Bandreject (notch)

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**Drawbacks: phase distortion and ringing**

IIR Filter The feedback loop introduced creates the possibility of an infinite impulse (delayed sample). The simple averaging filter becomes an Exponential Time Averaging Filter (ETA Filter), equivalent to an infinitely long FIR filter. Care has to be taken with any feedback system. Feedback coefficients have to remain below 1.0, or the filter becomes unstable. IIR filters are computationally less expensive than FIR filters for greater shaping potential. Drawbacks: phase distortion and ringing

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**Design of IIR filters the feedback of previous outputs.**

Most recursive filters have an infinite impulse response, because of the feedback of previous outputs. Practical Infinite-Impulse-Response (IIR) filters are usually based upon analogue equivalents (Butterworth, Chebyshev, etc.), using a transformation known as the bilinear transformation which maps the s-plane poles and zeros of the analogue filter into the z-plane. However, it is quite possible to design an IIR filter without any reference to analogue designs, for example by choosing appropriate locations for the poles and zeroes on the unit circle

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Contd…

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