Time Domain Methods in Speech Processing General Synthesis ...
Time Domain Methods in Speech Processing General Synthesis ...
Time Domain Methods in Speech Processing General Synthesis ...
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Short-<strong>Time</strong> Energy, Magnitude, ZC<br />
Summary of Simple <strong>Time</strong> <strong>Doma<strong>in</strong></strong><br />
Measures<br />
xˆ ( n)<br />
L<strong>in</strong>ear x(n) T[x(n)] Lowpass ˆn<br />
T[ ]<br />
Filter<br />
Filter, w(n)<br />
∞<br />
∑<br />
Qnˆ= T( x[ m]) w[ nˆ−m] m=−∞<br />
1. Energy:<br />
Enˆ= nˆ<br />
∑<br />
m= nˆ− L+<br />
1<br />
2<br />
x [ m] w[ nˆ−m] i can downsample Enˆ<br />
at rate commensurate with w<strong>in</strong>dow bandwidth<br />
2. Magnitude:<br />
Mnˆ= nˆ<br />
∑<br />
m= nˆ− L+<br />
1<br />
x[ m] w[ nˆ−m] 3. Zero Cross<strong>in</strong>g Rate:<br />
1<br />
Znˆ= z1= ∑ sgn( x − −<br />
2L<br />
= − +<br />
= ≥<br />
=− <<br />
ˆ n<br />
[ m]) sgn( x[ m 1])<br />
m nˆ L 1<br />
where sgn( xm [ ]) 1 xm [ ] 0<br />
1 xm [ ] 0<br />
Periodic Signals<br />
• for a periodic signal we have (at least <strong>in</strong><br />
theory) Φ[P]=Φ[0] so the period of a<br />
periodic signal can be estimated as the<br />
first non-zero maximum of Φ[k]<br />
– this means that the autocorrelation function is<br />
a good candidate for speech pitch detection<br />
algorithms<br />
– it also means that we need a good way of<br />
measur<strong>in</strong>g the short-time autocorrelation<br />
function for speech signals<br />
Q<br />
Issues <strong>in</strong> ZC Rate Computation<br />
• for zero cross<strong>in</strong>g rate to be accurate, need zero<br />
DC <strong>in</strong> signal => need to remove offsets, hum,<br />
noise => use bandpass filter to elim<strong>in</strong>ate DC and<br />
hum<br />
• can quantize the signal to 1-bit for computation<br />
of ZC rate<br />
• can apply the concept of ZC rate to bandpass<br />
filtered speech to give a ‘crude’ spectral estimate<br />
<strong>in</strong> narrow bands of speech (k<strong>in</strong>d of gives an<br />
estimate of the strongest frequency <strong>in</strong> each<br />
narrow band of speech)<br />
43 44<br />
45<br />
47<br />
Short-<strong>Time</strong> Autocorrelation<br />
-for a determ<strong>in</strong>istic signal, the autocorrelation function is def<strong>in</strong>ed as:<br />
∞<br />
∑<br />
Φ[ k] = x[ m] x[ m+ k]<br />
m=−∞<br />
-for a random or periodic signal, the autocorrelation function is:<br />
L 1<br />
Φ[ k] = lim ∑ x[ m] x[ m+ k]<br />
N→∞<br />
( 2L+ 1)<br />
m=−L - if x[<br />
n] = x[ n+ P], then Φ[ k] = Φ[ k + P],<br />
=> the autocorrelation function<br />
preserves periodicity<br />
-properties of Φ[ k]<br />
:<br />
1.<br />
Φ[ k] is even, Φ[ k] = Φ[ −k]<br />
2. Φ[ k] is maximum at k = 0, | Φ[ k] | ≤Φ[ 0],<br />
∀k<br />
3. Φ[ 0]<br />
is the signal energy or power<br />
(for random signals)<br />
Short-<strong>Time</strong> Autocorrelation<br />
- a reasonable def<strong>in</strong>ition for the short-time autocorrelation is:<br />
∞<br />
R ˆ ˆ<br />
nˆ<br />
[ k] = ∑ x[ mw ] [ n− m] x[ m+ k] w[ n−k −m]<br />
m=−∞<br />
1. select a segment of speech by w<strong>in</strong>dow<strong>in</strong>g<br />
2. compute determ<strong>in</strong>istic autocorrelation of the w<strong>in</strong>dowed speech<br />
Rnˆ[ k] = Rnˆ[ −k]<br />
- symmetry<br />
∞<br />
= ∑ xmxm [ ] [ −k] ⎡⎣wn [ ˆ− mwn ] [ ˆ+<br />
k−m] ⎤⎦<br />
m=−∞<br />
- def<strong>in</strong>e filter of the form<br />
h [ ˆ] = [ ˆ] [ ˆ<br />
k n w n w n+ k]<br />
- this enables us to write the short-time autocorrelation <strong>in</strong> the form:<br />
∞<br />
R ˆ<br />
nˆ<br />
[ k] = ∑ xmxm [ ] [ −k] hk[ n−m] m=−∞<br />
th<br />
- the value of R ˆ<br />
nˆ<br />
[ k] at time n for the k lag is obta<strong>in</strong>ed by filter<strong>in</strong>g<br />
the sequence xn [ ˆ] xn [ ˆ − k] with a filter with impulse<br />
response h [ ˆ k n]<br />
46<br />
48<br />
8