Continuous Fourier transform of a Gaussian Function

Transcription

1 Continuous Fourier transform of a Gaussian Function Gaussian function: e t2 /(2σ 2 ) The CFT of a Gaussian function is also a Gaussian function (i.e., time domain is Gaussian, then the frequency domain is also Gaussian: e t2 /(2σ 2 ) CFT Pair σ 2πe σ2 ω 2 /2

2 Uncertainty Principle of CFT For a signal x t, let P x = x t 2 dt = 1 X jw 2 dw 2π (from Parseval s theorem, the equality holds) Define the centroid of x t to be C x = 1 t x t 2 dt P x (similar to the definition of mean of a probability distribution)

3 Uncertainty Principle of CFT (cont.) Likewise, define the centroid of X jw to be C X = 1 w X jw P x in the frequency domain. 2 dw

4 Uncertainty Principle of CFT (cont.) The effective width of a signal x t is defined as 1/2 W x = 1 t C 2 P x x(t) 2 dt x The effective width of the spectrum X jw is defined as 1/2 W X = 1 w C 2 P X X jw 2 dw x (they are similar to the definition of standard deviation of a probability distribution)

5 Uncertainty Principle of CFT (cont.) Uncertainty Principle: It an be proven that W x W X 0.5 for any CFT transform pair x t and X(jw). Uncertainty Principle The product of the effective width of a signal and the effective width of its continuous Fourier transform is not smaller than 0.5.

6 Uncertainty Principle of CFT (cont.) W x W X 0.5 When will the equality hold? i.e. What are the signals satisfying the property W x W X = 0.5 Gaussian functions are the only functions for making the equality holds in the uncertainty principle. x t = e t2 /(2σ 2), X jw = σ 2πe σ2 ω 2 /2

7 What is Spectrum in general? There are different kinds of Fourier transforms. What is the one that defines generally the concept of spectrum? Answer: The continuous Fourier transform (CFT) defines the spectrum in general. CFT pair F jw f t e jwt 1 dt f t F jw e 2 jwt dw

8 Recall: Sampling for Processing In DSP, we always have to sample continuous-time (analog) signals into discrete-time signals for processing. Sampling in time domain: remember that if we perform sampling on an analog signal x a, x ( t) x ( nt ) ( t nt ) s n a Sampling in frequency domain: the spectrum becomes the sum of infinite many shifted copies of the original spectrum, 1 2 r X s jw X a jw j T r T

9 Recall: Aliasing and Sampling Theorem Hence, if the analog signal is band-limited with the frequency bound w b : X a (jw) = 0 for w >w b, and the sampling rate satisfies the Nyquist sampling theorem that w s > 2w b. Then, we know that the aliasing effect can be avoided, and the analog signal x a can be reconstructed by applying an ideal low-pass filter with the cutoff frequency w s.

10 Recall: Why using DTFT Since we have always to deal with discrete-time signals in DSP, we have defined a Fourier transform, DTFT, particularly for discrete-time signal processing. DTFT pair: X jw e x n n e jwn 1 2 jw n X e The spectrum of DTFT is defined in [-, ], (implicitly assume that the discrete-time signals are sampled satisfying the Nyquist rate). x e jwn dw

11 Recall: How DTFT approximates CFT? Exact recovery: when the X a (jw) is band-limited and the sampling rate is high enough such that the sampling theorem is satisfied, X a (jw) can be exactly recovered from the DTFT X(e jwt ) (by investigating the range only in w [- /T, /T ]). Approximation: When the sampling rate is not high enough or X a (jw) is not band-limited, X(e jwt ) is only an approximation of X a (jw) because of the aliasing effect (some high-frequency part will be folded to the range [- /T, /T ]). How DTFT approximates CFT can be completely characterized by sampling theorem and aliasing effect.

12 How to compute the spectrum? After converting an analog signal to discrete-time samples, another practical problem is how to compute the CFT spectrum (so that we can transform the signal to the frequency domain). Although DTFT can be used to recover the exact spectrum for band-limited signal under high-enough sampling rate, it requires summing from n= to. This is still infeasible in practice since we cannot compute the sum for an infinite-long signal.

13 Approximation by Finite-duration Signals So, what can we do? A practical way commonly employed is to use a finite range t [-T/2, T/2] of the analog signal x a (t), and see how it can approximate the spectrum of the entire signal defined in t (, ). After sampling with x[n]=x a (nt), there are N=T/T samples in the range t [-T/2, T/2], resulting a finite-duration discrete-time signal y[n] from x[n], y[ n] x[ n], 0, 0 n N 1 otherwise

14 Approximation by Finite-duration Signals Compute the DTFT for the finite-duration signal y[n] (now it is feasible in practice), and see how it can approximate the DTFT of x[n]. We can expect that, the larger is N (or equivalently, the larger is the range T), the better is the approximated spectrum.

15 Approximation by Finite-duration Signals In the above, there are two main factors affecting the approximation: (1) sampling, and (2) applying only a finite range of the signal. We have already seen how it approximates the waveform in the spectral domain by applying sampling (in time domain). Now we focus on the other factor: what is the approximation if we employ only a finite range of the signal?

16 Rectangular Windowing Employing a finite range t [T/2, T/2] of the analog signal x a (t), is equivalent to multiplying the original signal x a (t) with a rectangular window: 1, t [ T / 2, T / 2] w R ( t) 0, otherwise Time domain multiplication Frequency domain convolution (up to a scale) So, in the frequency domain, the spectrum X a (jw) is convolved with the CFT W R (jw) that is a sync function

17 Recall: Basic CFT Properties Note that there is a scale ½.

18 Multiplication with a rectangular window in the analog domain Multiplication with the rectangular window, w R (t) Convolution with the following sync function then divided by 2,

19 Convolution with Sync Function What is the effect of convolution with a sync function? Note that when T (that is, w R (t) 1), the sync function approaches to the delta function 2 (w). w Convolution with a delta function and the divided by 2 recovers exactly the original spectrum.

20 Convolution with Sync Function In general, when the window is wider, the sync function is narrower, and vice versa. It is easy to realize that convolution with a narrower sync function approximates the original spectrum better.

21 Example: approximation for a single sinusoid Assume that there is a single sinusoidal signal applied by the rectangular window in time domain: Let us consider the magnitude response of the sync function:

22 Example: approximation for a single sinusoid Then, convolution of a spectrum of a single sinusoid with the sync function looks like

23 Approximation by rectangular windowing in the Analog domain It can be seen that, by using a finite-length portion of an analog signal, the approximation can be analyzed by convolving a sync function in the frequency domain. This convolution disturbs the original spectrum: the spectrum is blurred and somewhat oscillated. The narrower is the sync function (i.e., the longer is the window), the more accurate is the approximation.

24 Approximating the Spectrum DTFT of a finite-length portion of the signal In sum, when applying DTFT only for the samples in a rectangular window of the signal, the approximation is caused by (1) frequency domain convolution with a sync function (2) aliasing Although imperfect, this is a feasible way we can do in practice for finding the spectrum of a signal.

25 Approximating the Spectrum DTFT of a finite-length portion of the signal What can be done to further improve the performance? The rectangular window is applied in the above. It can be replaced by other better windowing functions (eg., Hann, Hamming) that will distort the shape of spectrum less. will be investigated in the future

26 Spectrogram In the above, we only use a single segment of the signal and compute its DTFT. It is natural to extend this idea by separating a signal into multiple segments in time, and compute the DTFT for each segment along the time axis. In this way, we can obtain a two-dimensional map, where the horizontal axis is time index of the segment, and the vertical axis is the frequency in [- /T, /T]. Spectrogram is the common way for viewing the spectral domain response for a signal in practice.

27 Spectrogram Examples Synthesized notes, C,D,E,F,G,A,B,C Spectrogram of Fur Elise played on a piano

28 Spectrogram Examples