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**Systems: Definition Filter**

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

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**Systems and Properties: Linearity**

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**Systems and Properties: Time Invariance**

if S time then

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**Systems and Properties: Stability**

Bounded Input Bounded Output

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**Systems and Properties: Causality**

the effect comes after the cause. Examples: Causal Non Causal

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**Finite Impulse Response (FIR) Filters**

General response of a Linear Filter is Convolution: Written more explicitly: Filter Coefficients

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**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.

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**FIR Filter Response to an Exponential**

Let the input be a complex exponential Then the output is Filter

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Example Filter Consider the filter with input Then and the output

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**Frequency Response of an FIR Filter**

is the Frequency Response of the Filter

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**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:

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**Example Consider the Filter Filter defined as Let the input be:**

Expand in terms of complex exponentials:

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

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**The Discrete Time Fourier Transform (DTFT)**

Given a signal of infinite duration with define the DTFT and the Inverse DTFT Periodic with period

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**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

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**Example: DTFT of a rectangular pulse …**

Consider a rectangular pulse of length N Then where

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**Example of DTFT (continued)**

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**Why this is Important Filter Recall from the DTFT Then the output**

Which Implies

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**Summary Linear FIR Filter and Freq. Resp.**

Filter Definition: Frequency Response: DTFT of output

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**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.

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**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.

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**Example of Frequency Response (continued)**

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