Question

How to design an FIR digital filter for a speech enhancement applicatio.
please provide calculations and justify the values of parameters used in the system.

Answer 1

FIR filters have the following primary advantages:

• They can have exactly linear phase.

• They are always stable.

• The design methods are generally linear.

• They can be realized efficiently in hardware.

• The filter startup transients have finite duration.

The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter.

FIR Filter Design by Windowing

Let's assume that we want to design a Finite Impulse Response (FIR) filter with the desired frequency response

Hd(ω)Hd(ω)

. The window method applies the inverse discrete-time Fourier transform to find the corresponding time-domain representation,

hd(n)hd(n)

, as given by

hd(n)=12π∫π−πHd(ω)ejnωdωhd(n)=12π∫−ππHd(ω)ejnωdω

Equation 1

Generally,

hd(n)hd(n)

is not of finite length and since we were looking for a finite impulse response, we can truncate

hd(n)hd(n)

and achieve an approximation of the desired response. This is equivalent to multiplying

hd(n)hd(n)

by a rectangular window.

As discussed in a previous article, From Filter Specs to Window Parameters in FIR Filter Design, we can use other window functions to achieve a trade-off between ripples in the passband and the sharpness of the transition band.

For many classic filters, we can easily calculate the integral of Equation 1. For example, assume that we are designing an ideal lowpass filter with cutoff frequency of

ωcωc

. Then, we have

Hd(ω)={10|ω|≤ωcelseHd(ω)={1|ω|≤ωc0else

Equation 2

In this particular example, some mathematical manipulations can easily lead us to the corresponding

hd(n)hd(n)

(see Equation (8) in this article) because

Hd(ω)Hd(ω)

is given by a simple equation. Then, we can apply a suitable window function to arrive at a finite-length response which is an approximation of the desired filter. For more details of this discussion, please see this article on FIR filter design by windowing.

However, if

Hd(ω)Hd(ω)

has a complicated equation, we wouldn’t be able to easily calculate Equation 1. In these cases, we can use the frequency sampling method to design FIR filters with an arbitrary

Hd(ω)Hd(ω)

.

In this article, we will first review a practical example where the required FIR filter has a complicated frequency response and the frequency sampling method is quite helpful. Then, we will review the basics of FIR filter design using this method.

An FIR Filter to Compensate for the

sin(x)xsin(x)x

Distortion

We can think of a practical DAC as an ideal DAC, which produces an impulse train, followed by a zero-order hold block.

FIR filters are filters having a transfer function of a polynomial in z - and is an all-zero filter in the sense that the zeroes in the z-plane determine the frequency response magnitude characteristic. The z transform of a N-point FIR filter is given by :

FIR filters are particularly useful for applications where exact linear phase response is required. The FIR filter is generally implemented in a non-recursive way which guarantees a stable filter. FIR filter design essentially consists of two parts (i) approximation problem (ii) realization problem The approximation stage takes the specification and gives a transfer function through four steps. They are as follows: (i) A desired or ideal response is chosen, usually in the frequency domain. (ii) An allowed class of filters is chosen (e.g. the length N for a FIR filters). (iii) A measure of the quality of approximation is chosen. (iv) A method or algorithm is selected to find the best filter transfer function. The realization part deals with choosing the structure to implement the transfer function which may be in the form of circuit diagram or in the form of a program. There are essentially three well-known methods for FIR filter design namely: (1) The window method (2) The frequency sampling technique (3) Optimal filter design methods

The Frequency Sampling Technique In this method, [Park87], [Rab75], [Proakis00] the desired frequency response is provided as in the previous method. Now the given frequency response is sampled at a set of equally spaced frequencies to obtain N samples. Thus , sampling the continuous frequency response Hd(w) at N points essentially gives us the N-point DFT of Hd(2pnk/N). Thus by using the IDFT formula, the filter co-efficients can be calculated using the following formula

Now using the above N-point filter response, the continuous frequency response is calculated as an interpolation of the sampled frequency response. The approximation error would then be exactly zero at the sampling frequencies and would be finite in frequencies between them. The smoother the frequency response being approximated, the smaller will be the error of interpolation between the sample points. One way to reduce the error is to increase the number of frequency samples [Rab75]. The other way to improve the quality of approximation is to make a number of frequency samples specified as unconstrained variables. The values of these unconstrained variables are generally optimized by computer to minimize some simple function of the approximation error e.g. one might choose as unconstrained variables the frequency samples that lie in a transition band between two frequency bands in which the frequency response is specified e.g. in the band between the passband and the stopband of a low pass filter.