Frequency Domain Filtering

Perhaps a better way to eliminate noise in the NOISY dataset is to use Fourier transform filtering techniques. Noise is basically unwanted high-frequency content in sampled data. Applying a lowpass filter to the noisy data allows low-frequency components to remain unchanged while high-frequencies are smoothed or attenuated. Construct a filter function by entering the following step-by-step commands:

Create a floating point array using FINDGEN which sets each element to the value of its subscript and stores it in the variable Y by entering:

Y=[FINDGEN(100),FINDGEN(100)-100]

Next, make the last 99 elements of Y a mirror image of the first 99 elements:

Y[101:199]=REVERSE(Y[0:98])

Now, create a variable filter to hold the filter function based on Y:

filter=1.0/(1+(Y/40)^10)

Finally, plot:

IPLOT,FILTER

Figure 5-5: Constructing a Filter Function

Figure 5-5: Constructing a Filter Function

To filter data in the frequency domain, we multiply the Fourier transform of the data by the frequency response of a filter and then apply an inverse Fourier transform to return the data to the spatial domain.

Now we can use a lowpass filter on the NOISY dataset and store the filtered data in the variable lowpass by entering:

LOWPASS=FFT(FFT(NOISY,1)*filter,-1)

Then plot:

IPLOT, LOWPASS

Figure 5-6: Using a LOWPASS Filter

Figure 5-6: Using a LOWPASS Filter

The same filter function can be used as a high-pass filter (allowing only the high frequency or noise components through) by entering:

NOISY=ORIGINAL+((RANDOMU(SEED,200)-.5)/ 2)  
HIGHPASS=FFT(FFT(NOISY,1)*(1.0-filter),-1)

Then plot:

IPLOT, HIGHPASS

Figure 5-7: Using a Highpass Filter

Figure 5-7: Using a Highpass Filter