Math 345: Applied Mathematics VIII: The Wobble Problem

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1 Math 345: Applied Mathematics VIII: The Wobble Problem Marcus Pendergrass November 12, 2008 Old audio recordings often suffer from so-called wow and flutter distortion. On a tape player, for instance, variations in the speed with which the tape travels past the recording head introduces a pitch distortion in the recorded sound. The distortion introduced by high-frequency variations in the recording speed is known as flutter. Wow refers to the distortion induced by low-frequency variations in the recording speed. I ll use the term wobble to refer to either kind of distortion. Obviously we d like to correct for wobble whenever it is present. Unfortunately this is impossible in many cases. One case in which it is possible to correct for wobble is when there is a sinusoid of a known frequency in the original sound that was recorded. Because the sinusoid itself is distorted by the wobble, it provides a usable reference which makes it possible to correct for the wobble. 1 Modeling the Record/Playback Process In order to analyze wobble, we need a model of the record and playback process. This model should capture the essential features of the process that we need to analyze, while suppressing details that are irrelevant to our problem. To quote an old saw, the model should be as simple as possible, but no simpler. In the simplest terms, the recording process is a process in which data is written sequentially to some medium. Playback is the reverse process, in which data is read sequentially from the medium. The particular medium - be it spooled wire, audio tape, optical disk, computer memory, or whatever - is not important. What is important as far as wobble is concerned is that the rate at which data is written to the medium may not be constant. From this point of view, we can think of the recording medium as essentially a tape, or more abstractly, a number line. During recording, the recording head moves from position to position along the tape, writing a data value at each position. Thus, at each position p along the tape, a data value r(p) is recorded. This value r(p) is equal to the signal value during recording when the recording head was at position p. I will refer to the function r as the recording. The position p of the recording head in the medium during the record process is a function of time. Call this function the record function, and denote it by ψ r : p = ψ r (t). (1) So ψ r (t) is the position of the recording head at the given time during recording. During playback, we have to allow for the possibility that the head moves through the medium in a way that is different from the way it moved during recording. The function that describes the movement of the head during playback is called the playback function, and is denoted by ψ pb : Figure 1 on the next page summarizes the situation. p = ψ pb (t). (2) 1

2 record/playback head medium p = ψ(t) r(p) = Figure 1. Basic Record/Playback Model. The recording r(p) gives the signal value recorded at position p in the medium. During the recording process, the position of the record head is governed by the record function ψ = ψ r. During playback, the position is governed by a possibly different function ψ = ψ pb. There is one important assumption we will need to make concerning the record function ψ r. Think about what would happen if the recording head ever reversed direction during the record process. If the medium were re-writable, then the the recording head would begin over-writing previously recorded signal values. Thus, data would be lost. If the medium were not re-writable, all data coming in after the reversal of direction would be lost. In either case there would be no hope of recovering the original signal from the recording. Thus, we will assume that the recording head always moves in one direction only. What mathematical property of ψ r does this correspond to? For each position p, there is a unique time t when the recording head was at position p. Thus the record function p = ψ r (t) is one-to-one onto its range, and is therefore invertible. The inverse function t = ψr 1 (p) gives the time during the record process when the record head was at position p. 2 From Signal To Recording To Playback Let s(t) denote the signal that is recorded. What is the mathematical relationship between s and the recording r? The signal at time t is recorded in the medium at position p, where p = ψ r (t). Thus we have s(t) = r(p) = r(ψ r (t)), (3) or more succinctly s = rψ r, (4) where the juxtaposition of functions denotes composition: rψ r = r ψ r. The record function ψ r is invertible, so we can solve for r, yielding r = sψ 1 r. (5) Now consider playback. Let ŝ(t) denote the played-back signal. What is played back at time t is the value recorded at position p, where p = ψ pb (t) is now given by the playback function. Therefore ŝ(t) = r(p) = r(ψ pb (t)), (6) or more succinctly ŝ = rψ pb. (7) Equations (4) and (7) imply the Reciprocity Theorem, which says that so long as the record and playback functions are the same, the played back signal ŝ will be the same as the recorded signal s. Theorem 1 (Reciprocity). ŝ = s for all input signals s if and only if ψ r = ψ pb. The proof is an exercise. Equations (5) and (7) allow us to express the played back signal as a function of the input signal and the record and playback functions: ŝ = sψ 1 r ψ pb (8) 2

3 Example 1. Suppose the record head is speeding up during recording, according to the formula ψ r (t) = e ct, where c is a positive constant. Assuming that the playback function is constant velocity, ψ pb (t) = v 0 t, find expressions for the recording and the played back signal when the signal that was recorded was a sinusoid at frequency f 0. Solution. We have s(t) = sin(2πf 0 t). The recording is and the played back signal is r(p) = s(ψr 1 (p)) = s(ln(p/c)) = sin(2πf 0 ln(p/c)), ŝ(t) = s(ψr 1 (ψ pb (t))) = s(ψr 1 (v 0 t)) = s(ln(v 0 t/c)) = sin(2πf 0 ln(v 0 t/c)). The played back signal is a tone whose frequency is decreasing as time goes on. (See Exercise 6 below.) Exercise 1. Prove the Reciprocity Theorem. Exercise 2. Suppose that both the record and playback functions are ideal, constant velocity functions, but with different velocities: ψ r (t) = v 0 t, ψ pb (t) = v 1 t Express the played back signal ŝ in terms of the original signal s. Do the same for their Fourier transforms. Exercise 3. Suppose the record function is ideal (i.e. constant velocity), while the playback head speeds up at constant acceleration during playback, according to ψ pb (t) = 1 2 at2, where a is a positive constant. Show that the played back sound is a linear chirp, and find the chirp rate. Exercise 4. Suppose the record function is ideal (i.e. constant velocity), while the playback head speeds up during playback, according to ψ pb (t) = e ct, where c is a positive constant. Show that the played back sound is an exponential chirp, and find the chirp rate. Exercise 5. Suppose the record head speeds up during recording, with constant acceleration. ψ r (t) = 1 2 at2 Assuming an ideal constant velocity playback function, find the played back signal ŝ, given that the input signal was a sinusoid. 3

4 Exercise 6. Consider a signal of the form s(t) = sin(g(t)), where g is a monotonically increasing function. The instantaneous frequency of this signal at time t is defined to be (instantaneous frequency of sin(g(t)) at time t) = 1 2π g (t). Explain why this is the correct definition of instantaneous frequency. (Hint: make a reasonable definition of the average frequency of s from time t to time t + t, then take the limit as t goes to 0.) 3 Wobble Equation (4) above shows how to recover the original signal from the recording, provided that the record function ψ r is known: simply compose the recording r with ψ r : s = rψ r The Wobble Problem is to recover the original signal s from the recording r when the record function ψ r is unknown. As it stands, the Wobble Problem is clearly unsolvable. However, if some a priori information about the original signal is available, there is hope that s can be recovered from r, at least partially. For instance, if we knew that s contained a sinusoid at some frequency f 0, then that sinusoid could be used as a reference to track variations in the playback speed, and thus recover the playback function ψ r. Let s begin our analysis of the Wobble Problem by taking a closer look at the record and playback functions. First of all, we may assume that the playback function ψ pb is an ideal, constant velocity function: p = ψ pb (t) = v 0 t, (9) where p is position, and t is time. This assumption involves no real loss of generality, so long as the playback function is known. Now consider the record function. Obviously, we can t assume that it is ideal, but we can write it in a way that makes it easy to compare to the ideal: p = ψ r (t) = v 0 t + ɛ(t). (10) So here we are thinking of the record function as some perturbation of the playback function. The perturbation is ɛ(t) = p v 0 t, (11) which represents the positional error at time t during the record process: the difference between the actual position p and the ideal position v 0 t that we would have been at had the record function been equal to the playback function. Note that ɛ(t) is in units of length. Recall that the record function is invertible: t = ψr 1 (p) represents the time t during the record process when the record head was at position p. Again, we want to think of this inverse as a perturbed version of the inverse playback function ψ 1 pb (p) = p/v 0. So we write it as t = ψ 1 r (p) = p v 0 + w(p), (12) where w(p) = t p v 0 (13) represents the timing error at position p in the recording: the difference between the actual time t and the ideal time p/v 0 at which we would have been at position p had the record function been equal to the playback function. I m going to refer to w(p) as the wobble at position p. Note that the units of wobble are time. The wobble function w is really the key to the Wobble Problem. The problem with not knowing ψ r is that we don t know what sample time t is associated with a given position p in the recording. Equation (12) shows how to recover those sample times, provided that we know the wobble function. 4

5 Finally, I m going to make one more assumption, namely that playback velocity v 0 is equal to 1. This is really more of a convention than an assumption, because I can always change my units of length to make v 0 = 1 in the new units. Under this convention, the previous equations are: Ideal playback function: Record function: Positional error at time t: Inverse record function: Wobble at position p: p = ψ pb (t) = t, (14) p = ψ r (t) = t + ɛ(t). (15) ɛ(t) = ψ r (t) t = p t, (16) t = ψ 1 r (p) = p + w(p), (17) w(p) = ψ 1 r (p) p = t p (18) Equations (14) through (17) constitute the wobbly record/playback model that we will use from here on out. The next figure shows these quantities graphically. p = Ψ r (t) p = t w(p) ε(t) p (t,p) Figure 2. Wobble model. The record function p = ψ r (t) is in red, while the ideal playback function p = t is in blue. The wobble w(p) is given by the length of the indicated horizontal line, while the positional error ɛ(t) is the length of the indicated vertical line. Note that if p = ψ r (t), then equations (16) and (18) imply that ɛ(t) = w(p). In other words, ɛ(t) = w(t + ɛ(t)). From a physical point of view, it makes sense to assume that these model equations are all continuous and differentiable. Note that under this model, if s(t) is the input signal, then the recording r(p) and the played back signal ŝ(t) are given by and t r(p) = s(p + w(p)), (19) ŝ(t) = s(t + w(t)). (20) Exercise 7. Let the record function be p = ψ r (t) = t 2 /2, for t 0. Find the associated wobble function w(p) and the positional error function ɛ(t). 5

6 Exercise 8. Suppose the input signal for the recording is the complex tone s(t) = e 2πif0t+iθ. Assuming an ideal unit velocity playback function, express the played back signal ŝ in terms of the input signal and the wobble function w. Also, express the Fourier transform of ŝ in terms of the Fourier transform of s and the wobble. Exercise 9. Let the record model be ψ r (t) = t + δ sin(2πf w t), where δ > 0 is small enough so that ψ r is invertible. As usual, assume an ideal unit velocity playback function. Let the input signal be s(t) = sin(2πf 0 t). Use Matlab R to do the following: 1. Plot the record function ψ r (t). 2. Plot the positional error function ɛ(t) 3. Plot the wobble function w(p). 4. Compute the played back sound ŝ(t). 5. Plot S = Fs and Ŝ = Fŝ on the same set of axes. Use the following parameters for these plots: sampling rate f s = hertz, duration T = 3 seconds δ = 0.05, f w = 2 hertz f 0 = 880 hertz Comment on your results. Also, answer the following question: how small does δ have to be before the played back sound ŝ is indistinguishable by ear from the recorded signal s? 4 The Wobble Problem for a Complex Tone Suppose the signal to be recorded is a complex tone s(t) = e 2πif0t+iθ. How does wobble affect this signal? By equation (19) the recording r is given by r(p) = e 2πif0(p+w(p))+iθ = e 2πif0p+iθ e 2πif0w(p) = s(p)φ(p), where φ(p) = e 2πif0w(p). So a wobbly complex tone is the product of the original signal and this new function φ. For most signals s, the wobbly recording won t be nearly so simple to express mathematically. But if we can understand the effects of wobble in this simple case, maybe we can apply those insights to more complex signals later on. Taking the Fourier transform of the previous equation yields R(f) = e iθ Φ(f f 0 ), (21) where Φ = Fφ. (We used the Convolution Theorem here.). Thus if the frequency f 0 of the tone is known beforehand, we can recover Φ (and therefore the wobble) by shifting the Fourier transform of the recording to the left by f 0 units: R(f + f 0 ) = e iθ Φ(f) (22) Now the inverse Fourier transform followed by a complex logarithm give us ln(f 1 [R(f + f 0 )](p)) = i (θ + 2πf 0 w(p)) mod( π, π]. (23) 6

7 Here ln is the principle branch of the complex logarithm. Since the wobble function w is differentiable, it can be recovered from (23). In fact, to ease notation we can rewrite (23) as where ln c (c(p)) = i (θ + 2πf 0 w(p)), (24) c(p) = F 1 [R(f + f 0 )](p), (25) and the function ln c jumps from branch to branch of the complex logarithm consistently with the smoothness of its argument. From the previous two equations we have The left-hand side of (26) is the recovered wobble - denote it by w: ln c (c(p)) 2πif 0 = w(p) + θ 2πf 0. (26) w(p) = ln c(c(p)) 2πif 0. (27) The recovered wobble equals the actual wobble function w plus some unknown constant offset. The wobble-recovery process is illustrated in the next figure. a b c d Figure 4. Wobble recovery processing for a complex tone. a) The frequency domain representations of the original sinusoid (in red, f 0 = 880 hertz), and the distorted play-back signal (in blue). b) The shifted spectrum of the distorted play-back signal Ŝ. c) The inverse Fourier transform of the shifted signal from b. The real part is in green, while the imaginary part is in blue. This is the function c(p) from equation (25). d) The top graph is i ln(c(p)). Notice that the angles are all between π and π. But w is smooth, so we can unwrap the angles smoothly, as shown in the bottom graph. Dividing the values in the bottom graph by 2πf 0 gives the wobble function w(p), modulo a constant offset. 7

8 Note that equation (26) still contains an unknown offset that depends on the unknown phase angle θ. What is its effect? Recall that the wobble w(p) gives the timing error at position p: t = p + w(p), where t is the actual time during record associated with position p. The record times t given by the recovered wobble w will differ from these, but only by a constant amount: t = p + w(p) = t + θ 2πf 0. This will simply introduce a delay in the recovered recording. But all the wobble-induced distortion can now effectively eliminated, because we have recovered the record function (modulo the delay). For digital recordings, the sample times t (which are unevenly spaced) are used to resample the recording on an evenly spaced time grid, thus removing the wobble-induced distortion. The resampling is usually accomplished by some interpolation scheme, most simply by linear interpolation. Exercise 10. The file wobblytone.wav (available in our soundlibrary) contains a recording of a sinusoid at f 0 = 440 hertz which has been distorted by wobble. Write Matlab R code to recover the original tone from this distorted recording. Note: you will need to think about how to handle the fact that the real sinusoid is a sum of two complex tones, one at f 0 = 440 hertz, and the other at f 0 = 440 hertz. 5 The General Wobble Problem We saw in the last section that recovering from the wobble-induced distortion in a sinusoidal signal is relatively simple. The same ideas will work for more complex signals, provided that they contain an isolated sinusoid. The idea is simple in principle: 1. identify the frequency band that contains the isolated (and distorted) sinusoid; 2. bandpass-filter the signal down to this band (the filtered signal is a distorted complex tone); 3. recover the wobble function from the distorted tone as in the previous section; 4. use the recovered wobble function to eliminate the wobble distortion by resampling, as before. Note that item 1 above requires an estimate of the bandwidth over which the sinusoid has been spread by the distortion. Since the sinusoid s frequency is known, this bandwidth can usually be estimated from a graph of the Fourier transform of the distorted recording. Figure 5 on the next page illustrates the wobble recovery process for this more general class of signals. The performance of this algorithm is clearly dependent on the degree of isolation between the reference sinusoid and the remainder of the signal. The ideal case is that the reference sinusoid is completely separated from the remainder of the signal in frequency after the wobble-induced distortion. Because this distortion has the effect of smearing out frequencies, a sinusoid that is isolated from the remaining signal before distortion may not be so isolated after distortion. Even so, we would expect the wobble-recovery algorithm to perform reasonably well even in this case, provided that the post-distortion overlap between the reference sinusoid and the remaining signal is not too large. Simulation testing is needed to quantify the actual performance of the algorithm in these situations. Exercise 11. The file mysterysignal.wav (available in our soundlibrary) is a spoken-word recording between some famous characters. Unfortunately, the recording has been distorted by wobble. However, because of electrical interference issues in the microphone that was used, there was a pure tone at 60 hertz tone in the original signal that was recorded. Use the techniques we ve discussed to recover the original signal from the distorted recording. 8

9 a b c d Figure 5. Wobble recovery processing for a more complex signal. a) The frequency domain representation of the distorted signal is in blue. The distorted signal contains an isolated (and distorted) sinusoid at f 0 = 440 hertz. The corresponding complex tone is highlighted in green. This provides the reference necessary to remove the wobble-induced distortion. b) The distorted signal is band-pass filtered to remove everything but the complex tone, which is then shifted to the origin. Recovery of the wobble can now proceed as in Figure 4. c) The recovered wobble function. d) The restored recording, after the wobble distortion has been removed. Exercise 12. Devise a suite of simulations to test the performance of the wobble recovery algorithm. Implement these in Matlab R, and report your results. 9

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