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Digital Filters

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Time domain to Frequency Domain and vice versa

Filter Characteristics

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•When frequency low-pass filter increases above the cut-off frequency, the filter output drops at a constant rate.

• Similarly when the frequency at the input of a high-pass filter decreases below the cut-off frequency, the filter output also drops at a constant rate.

•The constant drop in filter output voltage per decade increase (x10) or decrease (/10) in frequency, is referred to as roll off.

•The amplitude response of a low pass filter is flat from DC to a point where it begins to roll off.

•The amplitude response is defined as the point where the amplitude has decreased by 3 dB, to 70.7% of its original amplitude.

•The region from at or near DC to the point where the amplitude is down 3 dB is defined as the passband of the filter.

•Similarly the region where the amplitude has dropped to 3dB to the point where the gain drops to zero is called the stopband.

•The amplitude of the filter at ten times the 3 dB frequency is attenuated a total of 20 dB for a one pole filter, and a total of 40 dB for a two pole filter. i.e 20 dB per decade (10 times frequency)

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As far as power is concerned

dB = 20 log (Pout/Pin)

and for voltage

dB – 10 log(Vout/Vin)

Decibels

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Filter Characteristics

Low Pass Filter

Band Pass Filter

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Digital filters are also characterised by their response to an impulse:

The impulse response is an indication of how long the filter takes to settle into a steady state: it is also an indication of the filter's stability - an impulse response that continues oscillating in the long term indicates the filter may be prone to instability.

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Filtering in the frequency domain

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Moving Average Filter• Probably the simplest of all

filters is known as the moving or running average filter which simply takes a number of samples and then averages them.

• The diagram on the right shows a sine wave with a spike on top of it. The spike is unwanted and it would be advantageous to remove it.

• One approach is to average the value on either side of the spike with the value of the spike. This would not eliminate the spike, but would reduce it. It can be achieved using the following algorithm.

g(n) = f(n – 1) + f(n) + f(n + 1)

3

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

2.500

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Series1

n f(n) f(n-1) + f(n) + f(n+1)/3 -1 0.000

0 0.000

1 0.383

2 0.707

3 0.924

4 2.000

5 0.924

6 0.707

7 0.383

8 0.000

9 -0.383

10 -0.707

11 -0.924

12 -1.000

13 -0.924

14 -0.707

15 -0.383

16 0.000

17 0.000

END

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n -2 -1 0 1 2 3 4 5 6

δ [n] 0 0 0 1 0 0 0 0 0 0 0

δ [n-2] 0 0 0 0 0 1 0 0 0 0 0

Delta Function

x[n] = 2 δ [n ] + 4 δ [n – 1] + 6 δ [n – 2] + 4 δ [n – 3] + 2 δ [n – 4]

n -2 -1 0 1 2 3 4 5 6

2 δ [n] 0 0 0 2 0 0 0 0 0 0 0

4 δ [n-1] 0 0 0 0 4 0 0 0 0 0 0

6 δ [n-2] 0 0 0 0 6 0 0 0 0 0

4 δ [n-3] 0 0 0 0 0 0 4 0 0 0 0

2 δ [n-4] 0 0 0 0 0 0 0 2 0 0 0

x δ n 0 0 0 2 4 6 4 2 0 0 0

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Definitions

• A linear system may be defined as one which obeys the Principle of Superposition, which may be stated as follows:

– If an input consisting the sum of a number of signals is applied to a linear system, then the output is the sum, or superposition, of the system’s responses to each signal considered separately.

• A time-invariant system is one whose properties do not vary with time. The only effect of a time-shift on an input signal to the system is a corresponding time-shift in its output.

• A causal system is one if the output signal depends only on present and/or previous values of the input. In other words all real time systems must be causal; but if data were stored and subsequently processed at a later date, it need not be causal.