L6: Short-time Fourier analysis and synthesis

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1 L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude This lecture is based on chapter 7 of [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 1

2 Recap from previous lectures Overview Discrete time Fourier transform (DTFT) Taking the expression of the Fourier transform X jω = x(t)e jωt dt, the DTFT can be derived by numerical integration X e jω = x n e jωn where x n = x nt S and ω = 2πF F S Discrete Fourier transform (DFT) The DFT is obtained by sampling the DTFT at N discrete frequencies ω k = 2πF s N, which yields the transform X k = N 1 n=0 x n e j2π N kn Introduction to Speech Processing Ricardo Gutierrez-Osuna 2

3 Why is another Fourier transform needed? The spectral content of speech changes over time (non stationary) As an example, formants change as a function of the spoken phonemes Applying the DFT over a long window does not reveal transitions in spectral content To avoid this issue, we apply the DFT over short periods of time For short enough windows, speech can be considered to be stationary Remember, though, that there is a time-frequency tradeoff here x(t) X(f) SFTF (Hz) time (sa.) frequency (Hz) time (frames) Introduction to Speech Processing Ricardo Gutierrez-Osuna 3

4 The short-time Fourier transform in a nutshell Define analysis window (e.g., 30ms narrowband, 5 ms wideband) Define the amount of overlap between windows (e.g., 30%) Define a windowing function (e.g., Hann, Gaussian) Generate windowed segments (multiply signal by windowing function) Apply the FFT to each windowed segment [Sethares, 2007] Introduction to Speech Processing Ricardo Gutierrez-Osuna 4

5 Windowing function STFT: Fourier analysis view To localize the speech signal in time, we define a windowing function w n, τ, which is generally tapered at its ends to avoid unnatural discontinuities in the speech segment Any window affects the spectral estimate computed on it The window is selected to trade off the width of its main lobe and attenuation of its side lobes The most common are the Hann and Hamming windows (raised cosines) w n, τ = cos w n, τ = cos 2π n τ N w 1 2π n τ N 1 Introduction to Speech Processing Ricardo Gutierrez-Osuna 5

6 Rectangular Hann Hamming Introduction to Speech Processing Ricardo Gutierrez-Osuna 6

7 Discrete-time Short-time Fourier transform The Fourier transform of the windowed speech waveform is defined as X n, ω = x m w n m e jωn m= where the sequence f n m = x m w n m is a short-time section of the speech signal x m at time n Discrete STFT By analogy with the DTFT/DFT, the discrete STFT is defined as X n, k = X n, ω 2π ω= N k The spectrogram we saw in previous lectures is a graphical display of the magnitude of the discrete STFT, generally in log scale S n, k = log X n, k 2 This can be thought of as a 2D plot of the relative energy content in frequency at different time locations Introduction to Speech Processing Ricardo Gutierrez-Osuna 7

8 For a long window w n, the result is the narrowband spectrogram, which exhibits the harmonic structure in the form of horizontal striations For a short window w n, the result is the wideband spectrogram, which exhibits periodic temporal structure in the form of vertical striations narrowband 5000 wideband x(t) SFTF (Hz) SFTF (Hz) time (sa.) time (frames) time (frames) Introduction to Speech Processing Ricardo Gutierrez-Osuna 8

9 STFT: filtering view The STFT can also be interpreted as a filtering operation In this case, the analysis window w n plays the role of the filter impulse response To illustrate this view, we fix the value of ω at ω 0, and rewrite X n, ω 0 = x m e jω0m w n m m= which can be interpreted as the convolution of the signal x n e jω 0n with the sequence w n : X n, ω 0 = x n e jω0n w n and the product x n e jω 0n can be interpreted as the modulation of x n up to frequency ω 0 (i.e., per the frequency shift property of the FT) Introduction to Speech Processing Ricardo Gutierrez-Osuna 9

10 [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 10

11 Alternatively, we can rearrange as (without proof) X n, ω 0 = e jω 0n x n w n e jω 0n In this case, the sequence x n is first passed through the same filter (with a linear phase factor e jω 0n ), and the filter output is demodulated by e jω 0n [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 11

12 This later rearrangement allows us to interpret the discrete STFT as the output of a filter bank X n, k = e j2π N kn x n w n e j2π N kn Note that each filter is acting as a bandpass filter centered around its selected frequency Thus, the discrete STFT can be viewed as a collection of sequences, each corresponding to the frequency components of x n falling within a particular frequency band This filtering view is shown in the next slide, both from the analysis side and from the synthesis (reconstruction) side Introduction to Speech Processing Ricardo Gutierrez-Osuna 12

13 analysis [Quatieri, 2002] synthesis Introduction to Speech Processing Ricardo Gutierrez-Osuna 13

14 Examples ex6p1.m Generate STFT using Matlab functions ex6p2.m Generate filterbank outputs using the filtering view of the STFT ex6p3.m Time-frequency resolution tradeoff (Quatieri fig 7.8) Introduction to Speech Processing Ricardo Gutierrez-Osuna 14

15 Short-time synthesis Under what conditions is the STFT invertible? The discrete-time STFT X n, ω is generally invertible Recall that X n, ω = f n m e jωn m= with f n m = x m w n m Evaluating f n [m] at m = n we obtain f n [n] = x n w 0 So assuming that w 0 0, we can estimate x n as x n = 1 X n, ω e jωn dω 2πw 0 This is known as a synthesis equation for the DT STFT π π Introduction to Speech Processing Ricardo Gutierrez-Osuna 15

16 Amplitude Redundancy of the discrete-time STFT There are many synthesis equations that map X n, ω uniquely to x n Therefore, the STFT is very redundant if we move the analysis window one sample at a time n = 1,2,3 For this reason, the STFT is generally computed by decimating over time, that is, at integer multiples n = L, 2L, 3L For large L, however, the DT STFT may become non-invertible As an example, assume that w n is nonzero over its length N w In this case, when L > N w, there are some samples of x n that are not included in the computation of X n, ω Thus, these samples can have arbitrary values yet yield the same X kl, ω Since X kl, ω is not uniquely defined, it is not invertible N w Unaccounted temporal samples L 2L 3L n Introduction to Speech Processing Ricardo Gutierrez-Osuna 16

17 Amplitude Likewise, the discrete STFT x n, k is not always invertible Consider the case where w n is band-limited with bandwidth B If the sampling interval 2π N is greater than B, some of the frequency components in x n do not pass through any of the filters of the STFT Thus, those frequency components can have any arbitrary values yet produce the same discrete STFT In consequence, depending on the frequency sampling resolution, the discrete STFT may become non invertible B Lost spectral region 2π N 2 2π N 3 2π N ω [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 17

18 Synthesis: filter bank summation FBS is based on the filtering interpretation of the STFT As we saw earlier, according to this interpretation the discrete STFT is considered to be the set of outputs from a bank of filters In the FSB method, the output of each filter is modulated with a complex exponential, and these outputs are summed to recover the original signal y n = 1 Nw 0 m= X n, k e j2π N nk [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 18

19 Under which conditions does FBS yield exact synthesis? It can be shown that y n = x n if either 1. The length of w n is less than or equal to the no. of filters N w > N, or 2. For N w > N: N 1 W ω 2π k = Nw 0 k=0 N The latter is known as the BFS constraint, and states that the frequency response of the analysis filters should sum to a constant across the entire bandwidth [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 19

20 Synthesis: Overlap-add OLA is based on the Fourier transform view of the STFT In the OLA method, we take the inverse DFT for each fixed time in the discrete STFT In principle, we could then divide by the analysis window This method is not used, however, as small perturbations in the STFT can become amplified in the estimated signal y n Instead, we perform an OLA operation between the sections This works provided that w n is designed such that the OLA effectively eliminates the analysis windows from the synthesized sequence The intuition is that the redundancy within overlapping segments and the averaging of the redundant samples averages out the effect of windowing Thus, the OLA method can be expressed as y n = 1 N 1 X p, k e j2π N kn W 0 k=0 p= where the term inside the square brackets is the IDFT Introduction to Speech Processing Ricardo Gutierrez-Osuna 20

21 Under which conditions does OLA yield exact synthesis? It can be shown that if the discrete STFT has been decimated by a factor L, the condition y n = x n is met when which holds when either p= w pl n = W 0 1. The analysis window has finite bandwidth with maximum frequency ω c less than 2π/L, or 2. The sum of all the analysis windows (obtained by sliding w n with L-point increments) adds up to a constant In this case, x n can then be resynthesized as x n = L W 0 p= 1 N N 1 k=0 L X pl, k e j2π N kn [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 21

22 [Quatieri, 2002] Introduction to Speech Processing Ricardo Gutierrez-Osuna 22

23 STFT magnitude The spectrogram (STFT magnitude) is widely used in speech For one, evidence suggests that the human ear extracts information strictly from a spectrogram representation of the speech signal Likewise, trained researchers can visually read spectrograms, which further indicates that the spectrogram retains most of the information in the speech signal (at least at the phonetic level) Hence, one may question whether the original signal x n can be recovered from X n, ω, that is, by ignoring phase information Inversion of the STFTM Several methods may be used to estimate x n from the STFTM Here we focus on a fairly intuitive least-squares approximation Introduction to Speech Processing Ricardo Gutierrez-Osuna 23

24 Least-squares estimation from the STFT magnitude In this approach, we seek to estimate a sequence x e n whose STFT magnitude X e n, ω is closest (in a least-squared-error sense) to the known STFT magnitude X n, ω The iteration takes place as follows An arbitrary sequence (usually white noise) is selected as the first estimate x e 1 n We then compute the STFT of x 1 e n and modify it by replacing its magnitude by that of X n, ω X 1 m, ω = X m, ω X e i m, ω i X e m, ω From this, we obtain a new signal estimate as i 1 x i m= w m n g m n e n = w 2 m n m= whereg m i 1 n is the inverse DFT of X i 1 m, ω And the process continues iteratively until convergence or a stopping criterion is met Introduction to Speech Processing Ricardo Gutierrez-Osuna 24

25 It can be shown that this process reduces the distance between X e n, ω and X n, ω at each iteration Thus, the process converges to a local minimum, though not necessarily a global minimum All steps in the iteration can be summarized as (Quatieri, 2002; p. 342) w m n 1 π x i+1 m= X e n = 2π i m, ω e jωn dω π w 2 m n m= where X i m, ω = X m, ω X e i m,ω X e i m,ω Introduction to Speech Processing Ricardo Gutierrez-Osuna 25

26 Example ex6p4.m Estimate a signal from its STFT magnitude Introduction to Speech Processing Ricardo Gutierrez-Osuna 26

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