## Matlab Filter Implementation

In this section, we will implement (in matlab) the simplest lowpass filter

- Fig.1.3 listed
`simplp`for filtering one block of data, and - Fig.1.4 listed a main program for testing
`simplp`.

`filter`

^{3.3}which will implement

`simplp`as a special case. The syntax is

y = filter (B, A, x)where

`x`is the input signal (a vector of any length),`y`is the output signal (returned equal in length to`x`),`A`is a vector of filter*feedback coefficients*, and`B`is a vector of filter*feedforward coefficients*.

`filter`function performs the following iteration over the elements of

`x`to implement any causal, finite-order, linear, time-invariant digital filter:

^{3.4}

where , , , is the length of

`B`, is the length of

`A`, and is assumed to be 1. (Otherwise,

`B`and

`A`are divided through by

`A`(1). Note that

`A`(1) is not used in Eq.(2.1).) The relatively awkward indexing in Eq.(2.1) is due to the fact that, in matlab, all array indices start at 1, not 0 as in most C programs.

Note that Eq.(2.1) could be written directly in matlab using two
`for` loops (as shown in Fig.3.2). However, this
would execute *much* slower because the matlab language is
*interpreted*, while built-in functions such as `filter`
are pre-compiled C modules. As a general rule, matlab programs should
avoid iterating over individual samples whenever possible. Instead,
whole signal vectors should be processed using expressions involving
vectors and matrices. In other words, algorithms should be
``vectorized'' as much as possible.
Accordingly, to get the most out of matlab, it is necessary to know
some linear algebra [58].

The simplest lowpass filter of Eq.(1.1) is *nonrecursive*
(no feedback), so the feedback coefficient vector `A` is set to
1.^{3.5} Recursive filters will be
introduced later in §5.1. The minus sign in
Eq.(2.1) will make sense after we study
*filter transfer functions* in Chapter 6.

The feedforward coefficients needed for the simplest lowpass filter are

With these settings, the `filter` function implements

% simplpm1.m - matlab main program implementing % the simplest lowpass filter: % % y(n) = x(n)+x(n-1)} N=10; % length of test input signal x = 1:N; % test input signal (integer ramp) B = [1,1]; % transfer function numerator A = 1; % transfer function denominator y = filter(B,A,x); for i=1:N disp(sprintf('x(%d)=%f\ty(%d)=%f',i,x(i),i,y(i))); end % Output: % octave:1> simplpm1 % x(1)=1.000000 y(1)=1.000000 % x(2)=2.000000 y(2)=3.000000 % x(3)=3.000000 y(3)=5.000000 % x(4)=4.000000 y(4)=7.000000 % x(5)=5.000000 y(5)=9.000000 % x(6)=6.000000 y(6)=11.000000 % x(7)=7.000000 y(7)=13.000000 % x(8)=8.000000 y(8)=15.000000 % x(9)=9.000000 y(9)=17.000000 % x(10)=10.000000 y(10)=19.000000 |

A main test program analogous to Fig.1.4 is shown in
Fig.2.1. Note that the input signal is processed in one big
block, rather than being broken up into two blocks as in Fig.1.4.
If we want to process a large sound file block by block, we need some
way to initialize the state of the filter for each block using the
final state of the filter from the preceding block. The
`filter` function accommodates this usage with an additional
optional input and output argument:

[y, Sf] = filter (B, A, x, Si)

`Si`denotes the filter

*initial*state, and

`Sf`denotes its

*final*state. A main program illustrating block-oriented processing is given in Fig.2.2.

% simplpm2.m - block-oriented version of simplpm1.m N=10; % length of test input signal NB=N/2; % block length x = 1:N; % test input signal B = [1,1]; % feedforward coefficients A = 1; % feedback coefficients (no-feedback case) [y1, Sf] = filter(B,A,x(1:NB)); % process block 1 y2 = filter(B,A,x(NB+1:N),Sf); % process block 2 for i=1:NB % print input and output for block 1 disp(sprintf('x(%d)=%f\ty(%d)=%f',i,x(i),i,y1(i))); end for i=NB+1:N % print input and output for block 2 disp(sprintf('x(%d)=%f\ty(%d)=%f',i,x(i),i,y2(i-NB))); end |

**Next Section:**

Simulated Sine-Wave Analysis in Matlab

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Elementary Filter Theory Problems