Non-Integer Time Shifting of Discrete Time Signal and Its Use in Beam Forming Denoising

 

Instructor: Professor Tufts

By Xu (Tony) Han

           

 

It is easy to do a time shifting for a continuous signal no mater what amount the time shift is. But when the continuous signal is sampled with some constant interval and thus is changed into discrete time signal, we can only know the signal value at some time instant, which is the multiple of the sample interval.

In this report,  is used to represent the continuous time signal and  is the corresponding discrete time signal to  for some sampling frequency .

                                                                      (1)

It is obviously very easy to shift  with some integer value denoted as m. We can simply get the shifted signal denoted as  through equation (2).

                                                                                              (2)

However, if we want to shift  with some non-integer value, how can we do? Because we only know the value of the discrete time signal at integer location, so the non-integer shifted signal can’t be got using equation (2). If we denote the non-integer shift as , we know that

                                 (3)

That is to say we want to know the value of the original continuous signal at some points, which are not sampled, from the discrete signal that is generated from sampling the continuous signal.

From Nyquist sampling theorem, we know that the continuous signal can be recovered from the discrete signal generated from sampling that continuous signal, if the sampling frequency is larger that the two times of the highest frequency of the continuous signal. Thus the value of the original continuous signal at some unsampled points definitely can be computed from the value of the sampled points if the condition of Nyquist theorem holds.

The problem that exists is how to compute the value of the unsampled points.

For a continuous signal, the CTFT (Continuous Time Fourier Transform) is:

                                                                                     (4)

                                                                               (5)

If the continuous signal is sampled by impulse train with the sampling frequency , (is the interval of sampled point), we know the corresponding sampled signal is:

                                                                       (7)

The impulse train and its Fourier transform is:

           

                                                                                     (8)

                                                 (9)

                                                                                                     (10)

                                              (11)

Also:

                                    (12)

            The DTFT (Discrete Time Fourier Transform) of the corresponding discrete time signal is:

           

                                                                    (13)

Review equation (12) and (13), if we replace the continuous time frequency  with discrete time frequency though the equation:

                                                                                                            (14)

The equation (12) and (13) are equal:

                                   (15)

            From equation (11) we know if the condition of Nyquist theorem holds,

                                                         (16)

           

 

 

 

 

                         (17)

Compare the equation (17) with the inverse DTFT of x[n]:

                                                                               (18)

We can made the conclusion that the value of the unsampled points can be computed using inverse CTFT (Equation 17) of the DTFT of x[n]. Thus we can get the non-integer time shifting of the discrete time signal from equation (3):

 

                             (19)

Let

                          (20)

*       

                                                                                                       (21)

Here the * is linear convolution.

Using Equation (21), we can realize the non-integer time shifting of discrete time signal. The result of shifting a sinusoidal impulse with 2.5 time units is given in figure 1.

 

 

Figure 1 Non-integer time delay result


            The conclusion we get from equation (21) about non-integer delay of discrete time signal can be used in beam forming denoising.

In beam forming denoising, we have an array of sensors, which pick up the signal contaminated by WGN (White Gaussian Noise) at different time start point.

Assume  is the contaminated discrete time signal pick up by the kth sensors, s(t) is the original continuous signal,  is the WGN in the kth sensor.

                                                                           (22)

Here d is the distance between two adjacent sensors; c is the spreading speed of the light and  is the arrival direction of the signal. We can shift  with non-integer value  and denote the shifted signal as ,

                                                                          (23)

Assume we have m sensors and the power of the WGN is ,

 Let                                       (24)

The SNR (Signal to Noise Ratio) of the kth sensor is:

 

 

 

 

                                                                               (25)

The SNR of the average of the delayed signal is:

                                              (26)

The SNR of the average of the delayed signal is improved by m times. That means it is a good method to denoising contaminated signal. Figure 2 shows the result of denoising contaminated sinusoidal impulse signal. 30 sensors are used in the experiment.

 

 

Figure 2  The result of denoising contaminated sinusoidal impulse signal by 30 sensors.

 

 

Conclusion:

            Non-integer time shifting of discrete time signal is very useful especially when we want to know the value of the signal between sampling points after A/D conversion. It also can be used in beam forming denoising and has good result.