Digital filter design – Introduction

Key focus: Develop basic understanding of digital filter design. Learn about fundamentals of FIR and IIR filters and the design choices.

Analog filters and digital filters are the two major classification of filters, depending on the type of signal signal they process. An analog filter, processes continuous-time signal analog signals. Whereas, digital filters process sampled, discrete-time signals.

Design of digital filters, involve the use of both frequency domain and time domain techniques.This is because, the filter specifications are often specified in frequency domain and the implementation is done in time-domain in the form of difference equations. Z-transform and discrete-time frequency transform (DTFT) are typical tools used for frequency domain analysis of filters.

Linear Time-Invariant Causal filter

The choice of filter and the design process depends on design specification, application and the performance issues associates with them. In this section, the focus is on the principles used for designing linear time-invariant causal filters.

A linear system obeys the principle of superposition. This means that an arbitrary signal can be represented as the weighted sums of shifted unit impulse functions. In a linear time-invariant filtering system, the filter coefficients do not change with time. Therefore, if the input signal is time-shifted, there will be a corresponding time-shift in the output signal. The term causal implies that the output of the system depends only on the present and past samples of input or output, and not on the future samples. Causality is required for real-time applications.

Solution for LTI causal filtering

The input-output relationship in a linear time-invariant causal filter system (shown above) is mathematically described using the following difference equation

where, x[n] and y[n] are the input and output (sampled discrete-time) signals of the filtering system, a_k and b_i are the coefficients of the filter that is programmed to certain values for achieving the given filtering task. The values of M and N determine the number of such coefficients, in other words, the filter has M zeros and N poles.

Linear time-invariant systems are completely characterized by it response to an impulse signal δ[n]. Therefore, the analysis and solution for a given filtering task, is easily achieved by using the impulse response h[n], which is the response of the LTI system to an impulse signal δ[n] .

Therefore, the solution for the recursive equation in [1] is found by exciting the input with an impulse , that is x[n]= δ[n].

Z-transform and DTFT

Z-transform and discrete-time frequency transform (DTFT) are important tools for analyzing difference equations and frequency response of filters. Z-transform converts a discrete-time signal into a complex frequency domain representation.

The impulse response of a discrete-time causal system is analyzed using the unilateral or one-sided Z-transform. The unilateral Z-transform of a discrete-time signal x(n) is given by

where, n is an integer and z is a complex number with magnitude r and complex argument ω in radians.

If we let |z| = r = 1, then the equation for Z-transform transforms into discrete-time Fourier transform.

The Z-plane contains all values of z, whereas, the unit circle (defined by |z|=r=1) contains only z=e ^jω values. Therefore, we can state that Z-transform may exist anywhere on the Z-plane and DTFT exists only on the unit circle. One period of DTFT is equivalent to one loop around the unit circle on the Z-plane.

One of the important properties of Z-transform is with regards to time-shifting of discrete-time samples. A delay of K samples in the time domain is equivalent to multiplication by z^-k in the Z-transform domain.

Re-writing equation (1) in Z-domain,

Therefore, the transfer function of the generic digital filter is given by

From equations (5) and (6), the frequency response of the system can be computed by evaluating the transfer function H(z) on the unit circle (i.e, |z| = r =1 → z = e^jω)

IIR filter

From implementation standpoint, there are two classes of digital filters: infinite impulse response (IIR) and finite impulse response (FIR) filters.

When a_k ≠ 0 as in equation (2), the filter structure is characterized by the presence of feedback elements. Due to the presence of feedback elements, the impulse response of the filter may not become zero beyond certain point in time, but continues indefinitely and hence the name infinite impulse response (IIR) filter.

Therefore, equation (9) and (10) are essentially the transfer function and the frequency response of an IIR filtering system.

There exist different methods for implementing the filter structure. Examples include, direct form I structure, direct form II structure, lattice structure, transposition, state space representation etc.., Each method has its own advantage and disadvantage. For example, some method may be robust to precision issues, coefficient sensitivity, lesser memory and computation requirement. The methods are chosen depending on the application.

Figure 2 shows the direct form I signal flow graph for the generic IIR filter described by equation (1). The boxes represented by Z^-1 are unit delays. This is because, the Z-transform of unit delay is Z^-1

FIR filter

When a_k=0 for all ks, there is no feedback in the filter structure, no poles in the pole-zero plot (in fact all the poles sit at the origin for a causal FIR). The equation is no longer recursive. The impulse response of such filter dies out (becomes zero) beyond certain point in time and hence the name finite impulse response (FIR) filter.

Setting x[n] = δ[n] and a_k=0 for all k, the impulse response of an FIR filter is given by,

Evaluating the z-transform of impulse input x[n] = δ[n] , in Z-domain following mapping holds

From equation (13) and (14), the transfer function of the FIR filter in Z-domain is given by

The frequency response is given by evaluating the transfer function on the unit circle |z|=1.

The direct form I signal flow graph for the FIR filter is shown in Figure 3

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References

[1] Z- transform – MIT open course ware ↗

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