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EEL 6586: HW#
Assignment is due Friday, March 9, 2007 in class. Late homework loses e#^ of^ days late^ − 1 percentage points. This assignment includes both Matlab and textbook questions.
Remember that the Exam will be periods E2-E3 on March 22, room TBA. Let the instructor know asap if you have a conflict
PART A: Short Answer (No more than a few sentences each)
A1 Compute the complex cepstrum of
H(z) = 1/(1 + az−^1 )
Assume |a| < 1.
A2 Compute the complex cepstrum of
H(z) = 1/(1 + az−^1 )^2
Assume |a| < 1.
A3 Compute the real cepstrum of
H(z) = 1/(1 + az−^1 )
Assume |a| < 1.
A4 Can we invert the Mel Frequency Cepstrum? Explain.
A5 Explain how cepstral mean subtraction can get rid of fixed convolu- tion artifacts due to various environmental and microphone transfer functions.
PART B: Textbook problems (Use Matlab only to optionally check your work)
B1 Compute the complex cepstrum of the following causal filter
H(z) =
1 + 18 z−^3
B2 Assume that a signal x(n) = (1/2)nu(n) is fed into the filter H(z) given in B1 to produce y(n). What is the complex cepstrum of y(n)?
B3 Compute the complex cepstrum of H(z) = 1 + az−^1 for |a| > 1 (non- minimum phase).
B4 Euclidean distance in complex cepstral space can be related to a RMS log spectral distance measure. Assuming that
log S(ω) =
n=+∑∞
n=−∞
cne−jnω
where S(ω) is the power spectrum (magnitude-squared Fourier trans- form), prove the following: n=+∑∞
n=−∞
(cn − c′ n)^2 =
2 π
∫ | log(S(ω)) − log(S′(ω))|^2 dω
where S(ω) and S′(ω) are the power spectra for two different signals.
B5 Assuming that
H(z) =
∑^ ∞
n=
h(n)z−n^ =
G
∑p k=1 a(k)z −k
Prove that the complex cepstrum hˆ(n) can be derived from the linear prediction coefficients a(k) using the following relation:
ˆh(n) = a(n) +
n∑− 1
k=
(k/n)ˆh(k)a(n − k)
for n ≥ 1.
PART C: Phoneme Recognition Experiments in Matlab
Utterances of 8 vowel phonemes from 38 speakers from the TIMIT database were extracted (about 2300 utterances total). Your goal for this problem is to achieve the highest recognition accuracy for this speech corpus. In the end you are free to do whatever you can to improve recognition accuracy. A demo Matlab program using LPC-10 and 1-NN is provided to demonstrate usage of the database. The following files are provided:
C4 For your final optimized system, which two phonemes are most likely to be confused with one another?
C5 Comment on why it is important that no speaker appear in both the test/train datasets.
As usual, attach all of your code to the end of the assignment. Bonus points will be awarded to the person(s) with the highest percentage correct classi- fication.
