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Systems: Definition Filter

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1 Systems: Definition Filter
A system is a transformation from an input signal into an output signal Example: a filter Filter SIGNAL NOISE

2 Systems and Properties: Linearity

3 Systems and Properties: Time Invariance
if S time then

4 Systems and Properties: Stability
Bounded Input Bounded Output

5 Systems and Properties: Causality
the effect comes after the cause. Examples: Causal Non Causal

6 Finite Impulse Response (FIR) Filters
General response of a Linear Filter is Convolution: Written more explicitly: Filter Coefficients

7 Example: Simple Averaging
Filter Each sample of the output is the average of the last ten samples of the input. It reduces the effect of noise by averaging.

8 FIR Filter Response to an Exponential
Let the input be a complex exponential Then the output is Filter

9 Example Filter Consider the filter with input Then and the output

10 Frequency Response of an FIR Filter
is the Frequency Response of the Filter

11 Significance of the Frequency Response
If the input signal is a sum of complex exponentials… Filter … the output is a sum is a sum of complex exponential. Each coefficient is multiplied by the corresponding frequency response:

12 Example Consider the Filter Filter defined as Let the input be:
Expand in terms of complex exponentials:

13 Example (continued) The frequency response of the filter is (use geometric sum) Then with Just do the algebra to obtain:

14 The Discrete Time Fourier Transform (DTFT)
Given a signal of infinite duration with define the DTFT and the Inverse DTFT Periodic with period

15 General Frequency Spectrum for a Discrete Time Signal
Since is periodic we consider only the frequencies in the interval If the signal is real, then

16 Example: DTFT of a rectangular pulse …
Consider a rectangular pulse of length N Then where

17 Example of DTFT (continued)

18 Why this is Important Filter Recall from the DTFT Then the output
Which Implies

19 Summary Linear FIR Filter and Freq. Resp.
Filter Definition: Frequency Response: DTFT of output

20 Frequency Response of the Filter
We can plot it as magnitude and phase. Usually the magnitude is in dB’s and the phase in radians.

21 Example of Frequency Response
Again consider FIR Filter The impulse response can be represented as a vector of length 10 Then use “freqz” in matlab freqz(h,1) to obtain the plot of magnitude and phase.

22 Example of Frequency Response (continued)

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