Filtering in the Frequency Domain

The smoothing operations discussed above have been presented in terms of the action of filters directly on the spectral data as recorded in the time domain. By converting the analytical spectrum to the frequency domain, the performance of these functions can be compared and a wide variety of other filters designed. Time-to-frequency conversion is accomplished using the Fourier transform. Its use was introduced earlier in this chapter in relation to sampling theory, and its application will be extended here. 

The electrical output signal from a conventional scanning spectrometer usually takes the form of an amplitude-time response, e.g. absorbance vs. wavelength. All such signals, no matter how complex, may be represented as a sum of sine and cosine waves. The continuous fimction of composite frequencies is called a Fourier integral. The conversion of amplitude-time, t, information into amplitude-frequency, w, information is known as a Fourier transformation. The relation between the two forms is given by 

The corresponding reverse, or inverse, transform, converting the complex frequency domain information back to the time domain is 

The two functions f t) and F w) are said to comprise Fourier transform pairs. 

As discussed previously with regard to sampling theory, real analytical signals are band-limited. The Fourier equations therefore should be modified for practical use as we cannot sample an infinite number of data points. With this practical constraint, the discrete forward complex transform is given by 

The smoothing operations discussed above have been presented in terms of the action of filters directly on the spectral data as recorded in the time domain. By converting the analytical spectrum into the frequency domain, the 

Time-to-frequency conversion is accomplished using the Fourier transform. Its use was introduced earlier in this chapter in relation to sampling theory, and its application will be extended here. 

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Instead of the application of a low pass filter in the frequency domain itself (multiplication with 1 in the low-frequency range and above that with 0, i.e. multiplication of the frequency response with a rectangle function), one can use in the amplitude domain (i.e. for the measurement itself) the mathematically fully equivalent convolution with the Fourier transform of that rectangle function. 

Numerically the convolution of a step scan is merely the application of a sliding weighted mean (e.g. like the Savitzky-Golay method). The Fourier transform of the rectangular function has the shape of sin(nv)/(nv) (whereby n is inversely proportional to the width of the rectangle) and unfortunately approaches 0 only very slowly. To make do with a small number of points for a convolution, one must tolerate a compromise and renounce the ideal rectangular shape of the low pass filter (in the frequency domain). 

Figure 7-4 Analyzing the First-order Low-pass RC Filter in the Frequency Domain...

Thus Tt/T is the number of pulse experiments in Tt. %ax is the maximum Fourier component of the signal after proper filtering. To obtain the maximum possible S/N ratio, the use of a matched filter is again necessary. The filtering is done in the time domain by multiplying the pulse response by exp(-t/T2). This corresponds to a matched filter in the frequency domain. 

When a signal y is corrupted with noise, filtering in the frequency domain can partly restore this signal into the deduced signal)) ... 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

The shaip cutoff at the limits -i and t, as illustrated by the boxcar function, often occurs in the frequency domain. In this case the boxcar acts as a low-pass filter in applications in electronics. All frequencies below [l] areunaltgEed, in this ideal case all higher ones are suppressed. 

Processing of time domain data may cause artefacts in the frequency domain. One example for these distortions are truncations at the beginning or at the end of the FID which could lead to severe baseline artefacts which can be reduced by an appropriate filter. Undesired resonances leading to broad lines in the final spectra can be more easily eliminated in time domain by truncating the first few data points. Furthermore, the model functions in time domain are mathematically simpler to handle than the frequency domain analogues, which leads to a reduction of computation time. The advantage of the frequency domain analysis is that the quantification process can be directly interpreted visually. 

The actual limit of the summation is the extent of the weighting filter. Zero padding is used to ensure that the discretized matrices have sizes which are a power of two so that the computation can be done in the frequency domain using fast fourier transform (FFT) techniques. The effective discretized density, pin, Wj), is then given by... 

The simplest way to sharpen up such data in the frequency domain is to use a Wiener filter (Press et al. 1986 Kino 1987). In the time domain, each... 

AutoFit Peaks III, Deconvolution method Deconvolution is a mathematical procedure that is used to remove the smearing or broadening of peaks arising because of the imperfection in an instrument s measuring system. Hidden peaks that display no maxima may do so once the data have been decon-voluted and filtered. This method requires a uniform x-spacing operated in the frequency domain. 

A very important consideration in this method is the location of the cutoff frequency for the transfer function. This point is the frequency where the transfer function changes from a value of nearly 1 to a value of nearly 0 frequencies above the cutoff frequency are mainly attenuated, while frequencies below this point are mainly passed through the filtering operation. In the example presented here, it was easy to see where the cutoff frequency should be placed, because signal and noise were well separated in the frequency domain. This convenient separation of signal and noise in the frequency domain is not always true, however. Noise and signal often overlap in the frequency domain as well as in the time domain. 

Figure 3 Sampling in the frequency domain (a) Modulated signal, x has frequency spectrum Xp (b) Harmonics of the carrier signal (c) Spectrum of modulated signal is a repetitive pattern of Xp and Xpcan be completely recovered by low pass filtering using, for example, a box filter with cut-offfrequency (d) Too low a...

Figure 4 Combining smoothing and differentiating in the frequency domain (a) the truncation filter to remove high frequency, noise sigruds and provide the first derivative (b) the transform of the spectrum form Figure 3(a) after application of the filter (c) the resultingfirst derivative spectrum from the inverse transform...

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