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Systems: Definition

S. Filter. SIGNAL. NOISE. Systems: Definition. A system is a transformation from an input signal into an output signal. Example: a filter. S. S. S. Systems and Properties: Linearity. Linearity:. S. time. time. S. time. time. Systems and Properties: Time Invariance.

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Systems: Definition

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

  2. S S S Systems and Properties: Linearity Linearity:

  3. S time time S time time Systems and Properties: Time Invariance if then

  4. Bounded Output Bounded Input S Systems and Properties: Stability

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

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

  7. Filter Example: Simple Averaging 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. Filter FIR Filter Response to an Exponential Let the input be a complex exponential Then the output is

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

  10. Frequency Response of an FIR Filter 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. Filter Example Consider the 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 periodicwe 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 Filter Definition: Frequency Response: DTFT of output

  20. Frequency Response of the Filter Filter Frequency Response: 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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