Answers to DSP Unit 2 & 3 Questions
1. Define pre-warping
Pre-warping is a technique used to compensate for the non-linear relationship between
analog and digital frequencies when converting analog filters to digital filters using methods
like bilinear transformation. It ensures that certain frequency points (usually cutoff
frequencies) are mapped correctly between the analog and digital domains.
2. Outline about signal flow graph
A signal flow graph is a graphical representation of a set of linear algebraic equations or a
digital system. It consists of nodes representing system variables and directed edges
representing functional relationships between the variables, making it a useful tool for
analyzing and understanding the signal paths in systems.
3. Analyze the digital transfer function H(z) by using impulse invariant
method for H(s) = 1/(s+2), assuming T=0.5 seconds.
The impulse invariant method involves sampling the impulse response of the analog system
and converting it into a discrete-time system. The transfer function would be:
H(z) = (1 - z^-1) / (1 - e^-2T z^-1)
For T = 0.5, this yields: H(z) = (1 - z^-1) / (1 - e^-1 z^-1).
4. Differentiate Hanning and Hamming windows
Hanning window: The Hanning window has smoother transitions at the ends, which
reduces the spectral leakage more than a rectangular window but less than a Hamming
window.
Hamming window: The Hamming window minimizes the maximum side lobe level, which
leads to better performance in terms of reducing spectral leakage, though its main lobe is
slightly wider than that of the Hanning window.
5. Outline the techniques for digitizing analog filters
Common techniques include:
- Impulse Invariant Transformation: Mapping the impulse response of an analog system
directly into the discrete domain.
- Bilinear Transformation: Warping the frequency axis to map the entire s-plane onto the z-
plane without aliasing.
- Matched Z-Transform: Mapping the poles and zeros of the analog filter to the digital
domain while preserving the filter's characteristics.
6. List the conditions for a digital filter to be causal and stable
Causality: A digital filter is causal if its output at any time depends only on present and past
inputs, not future ones.
Stability: A digital filter is stable if the absolute sum of the impulse response is finite, or if all
poles of the transfer function H(z) are inside the unit circle.
7. List the advantages and disadvantages of an FIR filter
Advantages:
- Always stable
- Linear phase response
- Can be easily designed using windowing techniques
Disadvantages:
- Requires higher filter order compared to IIR for a similar performance
- More computationally intensive due to higher order.
8. Sketch the frequency response of an N-point rectangular window
The frequency response of a rectangular window consists of a main lobe centered at zero
frequency and several side lobes, with the main lobe width inversely proportional to N. The
side lobes decrease in amplitude but exhibit significant leakage.
9. Find the direct form realization with the minimum number of
multipliers for H(z) = 1/2 + 1/4z^-1 + 1/4z^-2 + 1/2z^-3
This is a 3rd-order FIR filter. The direct form realization will have 4 taps with coefficients
1/2, 1/4, 1/4, 1/2. The minimum number of multipliers required is 4.
10. Mention the Butterworth polynomial for various orders
Order 1: H(s) = 1/(s+1)
Order 2: H(s) = 1/(s^2 + 1.414s + 1)
Order 3: H(s) = 1/(s^3 + 2s^2 + 2s + 1)
Higher orders follow the same pattern, with progressively more terms and increasing
coefficients.
11. What are the classifications of filters based on frequency response?
Low-pass filter: Passes low frequencies, attenuates high frequencies.
High-pass filter: Passes high frequencies, attenuates low frequencies.
Band-pass filter: Passes a band of frequencies, attenuates frequencies outside this band.
Band-stop filter: Attenuates a band of frequencies, passes frequencies outside this band.
All-pass filter: Passes all frequencies but alters the phase response.
12. Obtain the digital transfer function H(z) by using impulse invariant
method for H(s) = 1/(s+2), assuming T=1 second
Using the same steps as in question 3, for T = 1, the digital transfer function will be: H(z) =
(1 - z^-1) / (1 - e^-2 z^-1).
13. What conditions on the FIR sequence h(n) are imposed so that the
filter has linear phase?
For linear phase, the impulse response h(n) must be symmetric or anti-symmetric,
meaning:
h(n) = h(N-1-n)
for all n, where N is the length of the FIR filter.
14. Differentiate Hamming window with Kaiser window
Hamming window: Fixed window that reduces side lobes at the cost of slightly wider main
lobes.
Kaiser window: Parametric window that allows control over the trade-off between main
lobe width and side lobe attenuation.
15. How are phase and delay distortions introduced?
Phase and delay distortions are introduced when the phase response of a filter is non-linear.
This causes different frequency components to experience different delays, resulting in
signal distortion.
16. Explain the procedure for designing FIR filters using windows
1. Start with the desired ideal filter (e.g., ideal low-pass).
2. Truncate the impulse response of the ideal filter using a window function to limit the
filter length.
3. Adjust the window parameters to achieve the desired frequency response trade-offs
(main lobe width, side lobe level).
17. Draw the impulse response of an ideal low-pass filter
The impulse response of an ideal low-pass filter is a sinc function:
h(n) = sin(ωc n) / (π n)
where ωc is the cutoff frequency.
18. Design the direct form realization of a linear phase FIR system for N
even
A direct form realization for a linear phase FIR filter with even N has symmetric or anti-
symmetric coefficients, which can be implemented using a structure that minimizes
multiplications by reusing symmetric terms.
19. List the application of Kaiser window
Speech processing
Radar signal processing
Power spectral estimation
Seismic signal analysis
20. Mention the advantages and disadvantages of the rectangular
window technique
Advantages: Simple and easy to implement, requires minimal computation.
Disadvantages: Poor frequency resolution due to high spectral leakage from its
discontinuities.
21. Mention the application of Hamming windows in real-time
applications
Hamming windows are commonly used in:
- Speech signal processing
- Communication systems
- Spectral analysis.
22. Justify which windowing technique is commonly used and why
The Hamming window is commonly used because it offers a good balance between main
lobe width and side lobe attenuation, making it suitable for most signal processing
applications where reducing spectral leakage is important.
23. Draw the outline diagram of the cascade form
The cascade form consists of multiple second-order sections connected in series, where
each section realizes a portion of the total filter response.
24. Compare Finite and Infinite Impulse Response filters
FIR:
- Always stable
- Can have linear phase
- Requires higher order for sharp filters
IIR:
- May be unstable (poles outside unit circle)
- More computationally efficient (lower order for sharp filters)
- Generally non-linear phase response.
25. Sketch the outline diagram of the parallel form and list the need in
DSP
The parallel form breaks down a transfer function into parallel sections, each realizing a
part of the filter. This structure is often needed for numerical stability and efficient
implementation in DSP systems.
