lecture05: Filters

This lecture is concerned with filters, in particular LTI (Linear, Time-Invariant) filters, which we can write as a convolution. We also introduce the z-transforms and the analogy between filters and systems.

Topics covered

• Filters
• Possible filter properties
• Linear Filters
• Linear Time Invariant Filters
• Convolution
• Circular convolution
• Example
• Example DFT of circular convolution
• Linear Time Invariant Causal Filters
• Impulse response
• Memory
• FIR example: Moving Average
• FIR example: difference
• Example of IIR filter: EWMA
• Transfer function
• Types of filters
• Example: MA {\fontsize{14pt{14pt}\selectfont $ y(n) = \frac{1}{2N+1} \sum_{i=-N}^{N} x(n-i)$}}
• Example: difference {\fontsize{14pt{14pt}\selectfont $ y(n) = x(n) - x(n-1)$}}
• Example: EWMA {\fontsize{14pt{14pt}\selectfont $y(n) = a y(n-1) + (1-a) x(n) $}}
• What do they sound like?
• Why does {\tt He make your voice funny?}
• Financial data example
• Better filters
• Terminology
• Example filter comparison
• Filtering in the frequency domain
• Perfect filters and Gibb's phenomena
• Filtering in the time domain
• Z-transforms
• Z-transform and convolutions
• Inverse Z-transform
• More general IIR filters
• ARMA filters
• Example: EWMA
• Example: EWMA impulse response
• Filter invertibility
• Some simple filters
• Application: Dolby noise reduction
• An applications: denoising
• Example
• Example 2
• An applications: detect level changes
• Example
• Systems
• System properties
• Linear systems
• Linear time-invariant systems
• Non-linear systems
• Resonance
• Tacoma narrows bridge