filters - Digital Filters Effect such as echo and chorus...

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Digital Filters Effect such as echo and chorus are commonly referred to as temporal effects. They are easily understood as operations based of (relatively) large time delays. However, it's common that digital signals, such as digital audio, will be affected in the frequency domain. An application may need to emphasize certain frequencies or eliminate other frequencies. A simple example is the tone control on a media player. The bass control emphasizes or decrease the lower frequencies, while the treble control emphasized or decreases the higher frequencies. This type of operation is called digital filtering . Surprisingly simple blocks of code can modify the frequency response of a digital signal. This chapter describes the process of analyzing, then constructing, simple digital filters. Determining the Frequency Response of a Feedforward Digital Filter A feedforward filter is any filter than can be written in the form illustrated by Equation 1: = = N i i t i t x a y 0 1 The response characteristics of a feedforward digital filter can be determined using these five steps: Step 1: Determine the transfer function Step 2: Eliminate the negative exponents. Step 3: Factor the numerator, determining the zeros. Step 4: Plot the zeros of the numerator on the z-plane. Step 5: Compute the gain based on the distances from the unit circle to the zeros. Suppose you wish to know the frequency response of the filter 1 5 . 0 + = t t t x x y . This is the process: Step 1: Determine the transfer function. We determine that by simply converting the terms. The x t term becomes 1. The 0.5x t-1 term becomes 0.5z -1 . If you had a term -0.4x t-9 , that would become -0.4z -9 . z -1 means a time delay of 3 samples, which is the same as x t-1 . So, the transfer function for this filter is: 1 5 . 0 1 ) ( + = z z H .

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Step 2: Get rid of the negative exponents. We are going to treat the transfer equation as a polynomial. So, we want to get rid of the negative exponents. We do that by multiplying the equation by z/z: z z z z z z z H 5 . 0 5 . 0 1 5 . 0 1 ) ( 1 1 + = + = + = 2 Step 3: Factor the numerator, determining zeros. It's easy to see that the polynomial in the numerator of Equation 2 is -0.5. The zeros of the equation are any values that will cause the equation to have a value of zero. Step 4: Plot the zeros of the numerator on the z-plane. A zero is a complex number. This particular zero is a complex number if a real part of 5 . 0 and an imaginary part of zero. We can plot that on a unit circle called the z-plane . The plot for this particular filter is illustrated in Figure 1. Real Imaginary Z=-0.5 Figure 1 - z-plane plot of the zero -0.5. A z-plane is a 2D coordinate system with a unit circle . The two dimensions are real and imaginary . The X axis is the real axis and the Y axis is the imaginary axis. A unit circle is simply a circle with radius 1. We plot the zero by drawing a little circle at the coordinate -0.5. Note that this is a complex number. The real part is -0.5. The imaginary part is zero. So, this plots at the location shown in Figure 1.
Step 5: Determine the frequency response.

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filters - Digital Filters Effect such as echo and chorus...

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