Fir digital filters (Page 5/7)

Digital signal processing and Page 5 / 7

The Four Types of Linear Phase FIR Filters

Type 1.	The impulse response has an odd length and is even symmetric about its midpoint of $n = M = (N - 1) / 2$ which requires $h (n) = h (N - n - 1)$ and gives [link] and [link] .
Type 2.	The impulse response has an even length and is even symmetric about $M$ , but $M$ is not an integer. Therefore, there is no $h (n)$ at the point of symmetry, but it satisfies [link] and [link] .
Type 3.	The impulse response has an odd length as for Type 1 and has the odd symmetry of [link] , giving an imaginary multiplier for the linear-phase form in [link] with amplitude [link] .
Type 4.	The impulse response has an even length as for Type 2 and the odd symmetry of Type 3 in [link] and [link] with amplitude [link] .

Examples of the four types of linear-phase FIR filters with the symmetries for odd and even length are shown in [link] . Note that for $N$ odd and $h (n)$ odd symmetric, $h (M) = 0$ .

This figure consist of four separate graphs of four different types of FIR Filters. For all of these graphs the x axis is labeled Time Index, n, and the y axis is labeled Impulse response, h(n). The graphs consist of vertical lines extending from the x axis and ending in a hollow circles. The top left graph is labeled Type 1. FIR Filter. The vertical lines correspond to the numbers on the x axis 1-5. The ending points of the graph at at points (0,2), (1,3), (2,4), (3,3), and (4,2). The top right graph (Type 2) is similar except that the ending points are at (0,2), (1,3), (2,3) and (3,2). The bottom left graph (Type 3) similar with points at (0,2), (1,3), (2,0), (3,-3), and (4,-2). The bottom right graph is similar to the previous graph with points at (0,2), (1,3), (2,-3), and (-2,3). — Example of Impulse Responses for the Four Types of Linear Phase FIR Filters

For the analysis or design of linear-phase FIR filters, it is necessary to know the characteristics of $A (ω)$ . The most important characteristics are shown in [link] .

Characteristics of

A (ω)

for Linear Phase

TYPE 1.	Odd length, even symmetric $h (n)$
	$A (ω)$ is even about $ω = 0$	$A (ω) = A (- ω)$
	$A (ω)$ is even about $ω = π$	$A (π + ω) = A (π - ω)$
	$A (ω)$ is periodic with period = $2 π$	$A (ω + 2 π) = A (ω)$
TYPE 2.	Even length, even symmetric $h (n)$
	$A (ω)$ is even about $ω = 0$	$A (ω) = A (- ω)$
	$A (ω)$ is odd about $ω = π$	$A (π + ω) = - A (π - ω)$
	$A (ω)$ is periodic with period $4 π$	$A (ω + 4 π) = A (ω)$
TYPE 3.	Odd length, odd symmetric $h (n)$
	$A (ω)$ is odd about $ω = 0$	$A (ω) = - A (- ω)$
	$A (ω)$ is odd about $ω = π$	$A (π + ω) = - A (π - ω)$
	$A (ω)$ is periodic with period $= 2 π$	$A (ω + 2 π) = A (ω)$
TYPE 4.	Even length, odd symmetric $h (n)$
	$A (ω)$ is odd about $ω = 0$	$A (ω) = - A (- ω)$
	$A (ω)$ is even about $ω = π$	$A (π + ω) = A (π - ω)$
	$A (ω)$ is periodic with period $= 4 π$	$A (ω + 4 π) = A (ω)$

Examples of the amplitude function for odd and even length linear-phase filter $A (ω)$ are shown in [link] .

This figure consist of four graphs of four different type of FIR Filter Frequency Response over 2p. For all of these graphs, the x axis is labeled Normally Frequency and the y axis is labeled Amplitude Response, A. These graphs are arranged vertically. the top graph shows a wave form that has a consistent series of peaks and troughs. The peak is at about 6 and the bottom of the trough is at about 2. There are two complete wavelengths shown. The second graph is labeled Type 2. This graph has a differe waveform. The wave starts at (-2,-10) and follows a positive slope til it reaches the x axis at which point it levels out for a bit and then turns positive again. The wave peaks at (0,10) and the mirrors it rise with a negative slope with the same behavior. The third graph labeled Type 3 is similar to the first graph, except that the rise and fall of the waves are not equal. The rise to the peak of the wave is longer in duration than the fall to the bottom of the trough. The peak is at about 10 on the y axis and the bottom of the trough is at about -10. There are two complete wavelengths present. The bottom graph is labeled Type 4. It has a similar pattern to the second graph except that this one starts at the x axis and extends from peak to trough from 10 to -10 on the y axis. The wave descends from x axis with a negative slope then turns positive plateaus at the x axis and then continues its positive slope peaks and then descends. — Example of Amplitude Responses for the Four Types of Linear Phase FIR Filters

These characteristics reveal several inherent features that are extremely important to filter design. For Types 3 and 4, $A (0) = 0$ for any choice of filter coefficients $h (n)$ . This would not be desirable for a lowpass filter. Types 2 and 3 alwayshave $A (π) = 0$ which is not desirable for a highpass filter. In addition to the linear-phase characteristic that represents atime shift, Types 3 and 4 give a constant 90-degree phase shift, desirable for a differentiator or Hilbert transformer. The firststep in the design of a linear-phase FIR filter is the choice of the type most compatible with the specifications.

It is possible to uses the formulas to express the frequency response of a general complex or non-linear phase FIR filter by taking theeven and odd parts of $h (n)$ and calculating a real and imaginary “amplitude" that would be added to give the actual frequency response.

Calculation of fir filter frequency response

As shown earlier, $L$ equally spaced samples of $H (ω)$ are easily calculated for $L > N$ by appending $L - N$ zeros to $h (n)$ for a length-L DFT. This appears as

H (2 π k / L) = DFT {h (n)} for k = 0, 1, \dots, L - 1

This direct method of calculation is a straightforward and flexible approach. Only the samples of $H (ω)$ that are of interest need to be calculated. In fact, even nonuniform spacingof the frequency samples can be achieved by sampling the DTFT defined in [link] . The direct use of the DFT can be inefficient, and for linear-phase filters, it is $A (ω)$ , not $H (ω)$ , that is the most informative. In addition to the direct application of theDFT, special formulas are developed in Equation 5 from FIR Filter Design by Frequency Sampling or Interpolation for evaluating samples of $A (ω)$ that exploit the fact that $h (n)$ is real and has certain symmetries. For long filters, even these formulasare too inefficient, so the DFT is used, but implemented by a Fast Fourier Transform (FFT) algorithm.

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Source: OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6

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