Subject: Speech Processing

Topic: Speech Analysis in Time Domain

Difficulty: Low

(i) The second interpretation of the equation $$ x_s(e^{jw}) = \sum_{m= -\infty}^{\infty} x(m)w(n-m)e^{-jwn} $$ is linear filtering or convolution.

(ii) It is also evident from the equation that of every value of w, $ x_n(e^{jw}) $ is nothing but the convolution of the sequence w(n) with the sequence $ x(n)e^{-jwn} $

(iii) For any particular given value of w, $ x_n(e^{jw}) $ can be visualized as the output of a system. We can see that the output is complex.

(iv) Now, if $ x_n(e^{jw}) $ is expressed as, $ x_n(e^{jw}) = a_n(w) - jb_n(w) $

Linear Filtering using a) Complex operation b) Real operations only

(v) The method can be used to obtain a$_n$(w) and b$_n$(w) in which all sequences used are real.

(vi) When we consider n as variable and w as fixed, this can be considered as modulation low pass filtered from: $$ x(e^{jw}) = \sum_{m=-\infty}^{\infty} x(m)e^{-jwn} w(n-m) $$

(vii) The modulation process thus results in $ x(e^{j(\beta + w)}) $, which is the fourier transform of the input of the linear filter. The spectrum of the input sequence x(n) at the frequency w is shifted to zero frequency.

(viii) The output of the filter we have the fourier transform $ x(e^{j(\beta + w)})x(e^{j\beta}) $.

(ix) If the low pass filter has very narrow pass-band then the filter output would depend on $ x(e^{jw})$.

(x) This requires that the previous interpretation would require that $ w(e^{j \beta}) $ be non-zero for a very narrow band around zero frequency and must have small or negligible values outside this band.

(xi) It must be noted that the equation, $ \frac{1}{2 \pi} \int_{-\pi}^{\pi} w(e^{j \beta}) \times (e^{j(\beta + w)}) - e^{j \beta} \,\, d \beta $ is the inverse fourier transform of the output of the filter.

(xii) The other interpretation of $ x_n(e^{jw}) $ can be thought of as the result of $ e^{-jwn} $ being modulated with the output of a complex band-pass filter having impulse response $ w(n)e^{jwn} $

(xiii) If $ W(e^{j \beta}) $ is a low pass function then filter would be band-pass filter with a pass-band centered at frequency w.

(xiv) The band-pass filter demodulation can be written as: $$ X(e^{jwn}) = \sum_{m= -\infty}^{\infty} W[m] x[n-m]e^{-jw(n-m)} \\ X(e^{jwn}) = e^{-jwn} \sum_{m= -\infty}^{\infty} W[m]e^{jwm} x(n-m) \\ X(e^{jwn}) = e^{-jwn} [w[n]]e^{jwn} \times x[n] $$

Alternate method of linear filtering using complex operations.