1. Cepstrum and MFCC
Cepstrum
MFCC
Speech processing

2. Cepstrum
A new word by reversing the first 4 letters of spectrum  cepstrum.
It is the spectrum of a spectrum of a signal.
Speech processing

3. Cepstrum
Speech processing

4. Glottis and cepstrum Speech wave (X)= Excitation (E) . Filter (H)
Output
So voice has a
strong glottis
Excitation
Frequency content
In Ceptsrum
We can easily identify and
remove the glottal excitation
(H)
(Vocal tract filter)
(E)
Glottal excitation
From
Vocal cords
(Glottis)
Speech processing

5. Cepstral analysis Signal(s)=convolution(*) of
glottal excitation (e) and vocal_tract_filter (h)
s(n)=e(n)*h(n), n is time index
After Fourier transform FT: FT{s(n)}=FT{e(n)*h(n)}
Convolution(*) becomes multiplication (.)
n(time) w(frequency),
S(w) = E(w).H(w)
Find Magnitude of the spectrum
|S(w)| = |E(w)|.|H(w)|
log10 |S(w)|= log10{|E(w)|}+ log10{|H(w)|}
Ref:
Speech processing

6. Cepstrum C(n)=IDFT[log10 |S(w)|]=
IDFT[ log10{|E(w)|} + log10{|H(w)|} ]
In c(n), you can see E(n) and H(n) at two different positions
Application: useful for (i) glottal excitation (ii) vocal tract filter analysis
windowing
DFT
Log|x(w)|
IDFT
X(n)
X(w)
N=time index
w=frequency
I-DFT=Inverse-discrete Fourier transform
S(n)
C(n)
Speech processing

7. Cepstral for pitch detection
Dr. K.H. Wong, Introduction to Speech Processing
Cepstral for pitch detection
The theory behind the cepstral detector is that the fourier transform of a pitched signal usually have a number of regularly peaks, who is representing the harmonic spectrum.
When log magnitude of a spectrum is taken, these peaks are reduced (their amplitude brought into a usable scale).
The result is a periodic waveform in the frequency domain, where the period is related to the fundamental frequency of the original signal.
This means that a fourier transformation of this waveform has a peak representing the fundamental frequency.
Speech processing
V.74d

8. MFCC
MFCC is an efficient speech feature based on human hearing perceptions, i.e. MFCC is based on known variation of the human ear’s critical bandwidth with frequency.
Speech Signal
Pre-emphasis
Framing
Windowing Hamming
FFT
Mel Filter Bank
DCT
MFCC
Speech processing

9. Cont’d
Speech processing

10. MFCC
If x(n) is the input signal, then the short time Fourier transform for frame a is given
is called power spectrum, and if it is passed through triangular filters of Mel frequency filer bank
Speech processing

11. Cont’d
Human ear perception of frequency contents of sounds for speech signal does not follow a linear scale.
Therefore, for each tone with an actual frequency f, measured in Hz, a subjective pitch is measured on a scale called the MEL scale. The mel frequency scale is a linear frequency spacing below 1000 Hz and a logarithmic spacing above 1000Hz. To compute the mels for a given frequency f in Hz, a the following approximate formula is used.
Mel (f) = Sk = 2595*log10 (1 + f/700)
Speech processing

12. Cont’d
The subjective spectrum is simulated with the use of a filter bank, one filter for each desired mel-frequency component. The filter bank has a triangular band pass frequency response, and the spacing as well as the bandwidth is determined by a constant mel-frequency interval.
Furthermore, we convert the log mel spectrum back to time by using a discrete cosine transform (DCT) of the logarithm of S(m) is calculated to find the MFCC as
Speech processing

13. Filtering Ways to find the spectral envelope Filter banks: uniform
Filter banks can also be non-uniform
Spectral
envelop
spectral
envelop
energy
filter2
output
filter1
output
filter3
output
filter4
output
freq..
Speech processing

14. Filtering method
For each frame (ex ms), a set of filter outputs will be calculated. (ex frame overlap 5ms)
There are many different methods for setting the filter bandwidths -- uniform or non-uniform
Time frame i
Time frame i+1
Time frame i+2
Input waveform
30ms
Filter outputs (v1,v2,…)
Filter outputs (v’1,v’2,…)
Filter outputs (v’’1,v’’2,…)
Speech processing
5ms

15. How to determine filter band ranges
Uniform filter banks
Log frequency banks
Mel filter bands
Speech processing

16. Uniform Filter Banks Uniform filter banks
bandwidth B= Sampling Freq... (Fs)/no. of banks (N)
For example Fs=10Kz, N=20 then B=500Hz
Simple to implement but not too useful V
Filter
output v3 v1 v2 1 2 3 4 5
.... Q
...
freq..
(Hz)
500 1K
1.5K 2K
2.5K 3K
...
Speech processing

17. Non-uniform filter banks: Log frequency
Log. Freq... scale : close to human ear V
Filter
output v1 v2 v3
200
400
800
1600
3200
freq.. (Hz)
Speech processing

18. Inner ear and the cochlea (human also has filter bands)
Ear and cochlea
Speech processing

19. Mel filter bands (found by psychological and instrumentation experiments)
output
Freq. lower than 1 KHz has narrower bands (and in linear scale)
Higher frequencies have larger bands (and in log scale)
More filter below 1KHz
Less filters above 1KHz
Speech processing

20. Mel scale (Melody scale) From http://en. wikipedia
Measure relative strength in perception of different frequencies.
The mel scale, named by Stevens, Volkman and Newman in 1937 is a perceptual scale of pitches judged by listeners to be equal in distance from one another. The reference point between this scale and normal frequency measurement is defined by assigning a perceptual pitch of 1000 mels to a 1000Hz tone, 40 dB above the listener's threshold. …. The name mel comes from the word melody to indicate that the scale is based on pitch comparisons.
Speech processing

21. Critical band scale: Mel scale
Based on perceptual studies
Log. scale when freq. is above 1KHz
Linear scale when freq. is below 1KHz
Popular scales are the “Mel” (stands for melody) or “Bark” scales
Mel
Scale
(m) m
(f) Freq in hz f
Speech processing
Below 1KHz, fmf, linear
Above 1KHz, f>mf, log scale

22. Work examples:
Exercise 1: When the input frequency ranges from 200 to 800 Hz (f=600Hz), what is the delta Mel (m) in the Mel scale?
Exercise 2: When the input frequency ranges from 6000 to 7000 Hz (f=1000Hz), what is the delta Mel (m) in the Mel scale?
Speech processing

23. Work examples: Answer1: also m=600Hz, because it is a linear scale.
Answer 2: By observation, in the Mel scale diagram it is from 2600 to 2750, so delta Mel (m) in the Mel scale from 2600 to 2750, m=150 . It is a log scale change. We can re-calculate result using the formula M=2595 log10(1+f/700),
M_low=2595 log10(1+f_low/700)= 2595 log10(1+6000/700),
M_high=2595 log10(1+f_high/700)= 2595 log10(1+7000/700),
Delta_m(m) = M_high - M_low = (2595* log10(1+7000/700))-( 2595* log10(1+6000/700)) = (agrees with the observation, Mel scale is a log scale)
Speech processing

24. Example of cepstrum http://www. cse. cuhk. edu
Example of cepstrum Run spCepstrumDemo in matlab
'sor1.wav‘=sampling frequency 22.05KHz
Speech processing

25. (dft=discrete Fourier transform)
s(n) time
domain signal
x(n)=windowed(s(n))
Suppress two sides
|x(w)|=dft(x(n))
= frequency signal
(dft=discrete Fourier transform)
Log (|x(w)|)
C(n)=
iDft(Log (|x(w)|))
gives Cepstrum
Glottal excitation cepstrum
Vocal track
cepstrum
Speech processing

26. Liftering (to remove glottal excitation)
Low time liftering:
Magnify (or Inspect) the low time to find the vocal tract filter cepstrum
High time liftering:
Magnify (or Inspect) the high time to find the glottal excitation cepstrum (remove this part for speech recognition.
Vocal tract
Cepstrum
Used for
Speech
recognition
Glottal excitation
Cepstrum, useless for
speech recognition,
Cut-off Found
by experiment
Frequency =FS/ quefrency
FS=sample frequency
=22050
Speech processing

27. Reasons for liftering Cepstrum of speech
Why we need this?
Answer: remove the ripples
of the spectrum caused by
glottal excitation.
Too many ripples in the spectrum caused by vocal
cord vibrations (glottal excitation).
But we are more interested in
the speech envelope for recognition and reproduction
Fourier
Transform
Input speech signal x
Spectrum of x
Speech processing

28. Liftering method: Select the high time and low time liftering
Signal X
Cepstrum
Select high time, C_high
Select low time
C_low
Speech processing