ELT401 Digital Signal Processing - Lecture 6

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Questions and Answers

What is a primary characteristic of a Finite Impulse Response (FIR) filter?

  • It is an all-pole filter.
  • It consists only of feed forward components. (correct)
  • It has an infinite number of nonzero terms in its impulse response.
  • It always includes feedback components.

An IIR filter can be implemented without using any feedback components.

False (B)

What does 'H(z)' represent in the context of digital filters?

The transfer function of the filter.

In an IIR filter, the transfer function is given as H(z) = b0 / (1 + Σ ak z^−k). This indicates that it is primarily composed of ______ components.

<p>feedback</p> Signup and view all the answers

Match the following filter types with their characteristics:

<p>FIR Filter = Finite impulse response with feed forward characteristics IIR Filter = Infinite impulse response with feedback characteristics Transfer Function = Mathematical representation of the filter input-output relationship Signal Flow Graph = Visual representation of signal flow through system components</p> Signup and view all the answers

Which equation represents a linear constant-coefficient difference equation?

<p>y[n] = -Σ ak y[n − k] + Σ bl x[n − l] (B)</p> Signup and view all the answers

The Direct Form II implementation is designed to be less efficient than Direct Form I.

<p>False (B)</p> Signup and view all the answers

What is the purpose of developing efficient implementations in digital signal processing?

<p>To improve accuracy and reduce numerical precision requirements.</p> Signup and view all the answers

Which of the following statements about IIR filters is true?

<p>IIR filters can produce an output that depends on past inputs. (A)</p> Signup and view all the answers

FIR filters can have an infinite impulse response.

<p>False (B)</p> Signup and view all the answers

What are the primary advantages of IIR filters?

<p>Higher performance and efficient use of memory.</p> Signup and view all the answers

The output of a(n) ______ filter depends only on the current and past inputs.

<p>FIR</p> Signup and view all the answers

Which design technique is commonly used for selecting FIR filter coefficients?

<p>Hamming (D)</p> Signup and view all the answers

Match the following filter characteristics with their type:

<p>IIR = Feedback elements involved FIR = Linear phase response</p> Signup and view all the answers

IIR filters are generally used for applications requiring a linear phase response.

<p>False (B)</p> Signup and view all the answers

List one application where FIR filters are commonly used.

<p>Audio processing.</p> Signup and view all the answers

Flashcards

FIR Filter

A filter with a finite number of nonzero terms in its impulse response. Its transfer function has only zeros and its difference equation includes only feedforward terms.

IIR Filter

A filter with an infinite number of nonzero terms in its impulse response. Its transfer function has poles and zeros, and its difference equation includes feedback terms.

Signal Flow Graph

A representation of a filter's difference equation using a diagram with nodes and edges. Nodes represent signals, and edges represent operations (addition, multiplication, delay).

Direct Form I

A direct implementation of a filter's difference equation where each delay element (z^-1) is unique. This typically leads to more delay elements in the implementation.

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Direct Form II

A more efficient implementation of a filter's difference equation where delay elements are shared. This reduces the number of overall units needed.

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Difference Equation

A difference equation that describes the relationship between the input and output signals of a digital filter. It is a mathematical model of the filter's behavior. The equation includes coefficients that determine the filter's characteristics.

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Input Signal (x[n])

The input to the filter. This signal is transformed by the filter to produce the output.

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Output Signal (y[n])

The output of the filter. This signal is the result of the filter's operation on the input signal.

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How does an IIR filter's output depend on the input?

The output of an IIR filter depends on the current input as well as past inputs and outputs.

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How does an FIR filter's output depend on the input?

The output of an FIR filter only depends on the current and past inputs.

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Filter Design Techniques

A filter design technique where specific coefficients are chosen to create a desired frequency response, such as Butterworth, Chebyshev or Elliptic.

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Windowing Techniques

A filter design technique that uses windowing functions (e.g., Hamming, Hanning, Blackman) to create a filter with desired characteristics.

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Study Notes

ELT401 Digital Signal Processing - Lecture 6: FIR & IIR

  • Course is ELT401 Digital Signal Processing
  • Instructor is Dr. Nariman A. Khalil
  • Email is [email protected]

Digital Filters: Transfer Functions

  • Finding a filter's transfer function is about universal function approximation.
  • Common basis functions include polynomials and rational functions of z or z⁻¹.
  • Finite Impulse Response (FIR) filters are characterized by polynomial transfer functions.
  • Infinite Impulse Response (IIR) filters are characterized by rational function transfer functions.

FIR Filter

  • Transfer function (H(z)) = ∑(k=0 to M) bkz⁻ᵏ
  • bk are the numerator coefficients
  • M is the order of the filter

IIR Filter

  • Transfer function (H(z)) = ∑(k=0 to M) bkz⁻ᵏ/ ∑(k=0 to N) akz⁻ᵏ
  • bk are the numerator coefficients
  • ak are the denominator coefficients
  • M is the order of the numerator
  • N is the order of the denominator

Types of Filters

  • Feed-forward components filters have transfer function: y[n] = ∑(k=0 to M) bkx[n-k] and H(z) = Y(z)/X(z)= ∑(k=0 to M) bk z⁻ᵏ / 1

  • Feedback components filters have transfer function: y[n] = ∑(k=1 to N) aₖy[n-k] + ∑(k=0 to M) bₖx[n-k] and H(z) = ∑(k=0 to M) bₖ z⁻ᵏ / 1-∑(k=1 to N) aₖ z⁻ᵏ

Converting Difference Equations to Signal Flow Graphs

  • Linear, constant-coefficient difference equation: y[n] + a₁y[n-1] + a₂y[n-2] + ... + aₙy[n-N] = b₀x[n] + b₁x[n-1] + ... + bₘx[n-M]
  • Can write compactly as: y[n] = ∑(k=1 to N) aₖy[n-k] + ∑(k=0 to M) bₖx[n-k]
  • Can represent as a signal flow graph.

Direct Form II

  • More efficient implementation of a filter by sharing delay elements between the feedforward and feedback sections.
  • Has the same transfer function as Direct Form I.

Infinite Impulse Response (IIR) Filters

  • Impulse response continues indefinitely.
  • Output depends on current, past, and previous outputs.
  • High performance applications (audio, telecommunications).
  • Filter design typically involves choosing coefficients that multiply input and output signals.

Finite Impulse Response (FIR) Filters

  • Impulse response has a finite length.
  • Output depends only on current and past inputs.
  • Commonly used in applications like audio and image processing needing linear phase.
  • Filter design involves choosing filter length and coefficients (like Hamming, Hanning, Blackman windowing).

IIR vs. FIR Filters

  • IIR filters are typically more efficient (fewer taps) but can have non-linear phase.
  • FIR filters always have linear phase but are generally less efficient.

Choosing Between FIR and IIR Filters

  • FIR filters:

    • Exact linear phase.
    • Non-recursive implementations are typically stable.
    • Effects of limited bit-precision are less severe for IIR.
    • Require more coefficients for sharp cutoff filters.
    • Easier to synthesize arbitrary frequency responses.
    • Might need a Computer-aided design (CAD) support.

  • IIR filters:

    • Non-linear phase (especially at band edges)
    • Stability is not guaranteed.
    • Effects of limited bit-precision are often more severe.
    • Less coefficients resulting in less processing time.
    • Can readily convert analog filters to digital filters.
    • Easier to synthesize compared to FIR.

LTI System with Transfer Function & Direct Form II Implementation

  • One example of a signal processing method.

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