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### Definition of the Simplest Low-Pass

The simplest (and by no means ideal) low-pass filter is given by the following difference equation:

 (2.1)

where is the filter input amplitude at time (or sample) , and is the output amplitude at time . The signal flow graph (or simulation diagram) for this little filter is given in Fig.1.2. The symbol  '' means a delay of one sample, i.e., .

It is important when working with spectra to be able to convert time from sample-numbers, as in Eq.(1.1) above, to seconds. A more physical'' way of writing the filter equation is

where is the sampling interval in seconds. It is customary in digital signal processing to omit (set it to 1), but anytime you see an you can translate to seconds by thinking . Be careful with integer expressions, however, such as , which would be seconds, not . Further discussion of signal representation and notation appears in §A.1.

To further our appreciation of this example, let's write a computer subroutine to implement Eq.(1.1). In the computer, and are data arrays and is an array index. Since sound files may be larger than what the computer can hold in memory all at once, we typically process the data in blocks of some reasonable size. Therefore, the complete filtering operation consists of two loops, one within the other. The outer loop fills the input array and empties the output array , while the inner loop does the actual filtering of the array to produce . Let denote the block size (i.e., the number of samples to be processed on each iteration of the outer loop). In the C programming language, the inner loop of the subroutine might appear as shown in Fig.1.3. The outer loop might read something like fill from the input file,'' call simplp,'' and write out .''

 /* C function implementing the simplest lowpass: * * y(n) = x(n) + x(n-1) * */ double simplp (double *x, double *y, int M, double xm1) { int n; y[0] = x[0] + xm1; for (n=1; n < M ; n++) { y[n] = x[n] + x[n-1]; } return x[M-1]; } 

In this implementation, the first instance of is provided as the procedure argument xm1. That way, both and can have the same array bounds ( ). For convenience, the value of xm1 appropriate for the next call to simplp is returned as the procedure's value.

We may call xm1 the filter's state. It is the current memory'' of the filter upon calling simplp. Since this filter has only one sample of state, it is a first order filter. When a filter is applied to successive blocks of a signal, it is necessary to save the filter state after processing each block. The filter state after processing block is then the starting state for block .

Figure 1.4 illustrates a simple main program which calls simplp. The length 10 input signal x is processed in two blocks of length 5.

 /* C main program for testing simplp */ main() { double x[10] = {1,2,3,4,5,6,7,8,9,10}; double y[10]; int i; int N=10; int M=N/2; /* block size */ double xm1 = 0; xm1 = simplp(x, y, M, xm1); xm1 = simplp(&x[M], &y[M], M, xm1); for (i=0;i

You might suspect that since Eq.(1.1) is the simplest possible low-pass filter, it is also somehow the worst possible low-pass filter. How bad is it? In what sense is it bad? How do we even know it is a low-pass at all? The answers to these and related questions will become apparent when we find the frequency response of this filter.

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