Truncating Signals with Window Functions

2rn

Therefore for a signal with frequency <yo and duration rt to have its 1st zero crossing at midpoint of the ith side lobe of a signal with frequency coo and duration r0 , its duration

TI

must be such that:

2 n

+ ---= ^ m i d - p o i n t

= a + -2 Tn T; = '

2/ + 1

Therefore to target the ith side lobe using the fraction of the original data duration used is

*,= TT-T (4-10)

4.3.3 Truncating Signals with Window Functions

In the overview of window functions provided in section 4.2.3, it was shown how the use of window functions led to a loss in resolution through increased main lobe widths.

However, as the superimposition technique is based on altering the widths of the main lobe in signal spectra, this disadvantage is used to the benefit of the technique. By applying a window function to the data initially, it means that side lobes can be targeted using a

5 1

smaller truncation ratio with the additional benefit of side lobe suppression. The Hann, Hamming and Bartlet windows have zero crossing main lobe widths double that of the rectangular window which meansthat we could reduce theamount of data discarded by a half, whereas the Blackman windowon the other hand triplesthe zero crossing main lobe width thereby allowing us to reduce the amount of data discarded by a third.

For the Hann, Hamming and Bartlet windows that stretch the main lobe by a factor of 2, to target the i* side lobe using the combination of any of these windows, the data is truncated (before applying the window function) by :

* , = 7 ^ 7 (4' U)

For the Blackman window that stretches the main lobe by a factor of 3, to target the i* side lobe using the combination of any of this window, the data is truncated by:

*' = * 7 7 (412)

The windows considered were chosen due to their ease of implementation and this does not prevent the use of other windows in the technique as long as the ratio of the zero crossing main lobe width to rectangular window zero crossing main lobe width is correctly reflected when truncating data.

Figure 4.11 compares the superimposition spectrums where the 2nd side lobe is targeted using truncation and various windows. Figure 4.11 appears to show little difference in the level of side lobe suppression on the 2nd side lobe by the various windows. However, there is a difference in the other side lobes not targeted. As expected the other side lobes in the truncation-only spectrum are highest followed by the Bartlett, Hann and Hamming versions. Also, all but the Bartlett version have side lobes merged together.

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Truncation Only Version

— Hann Version Hamming Version

— Bartlett Version -1 0

2nd Lobe Position

C

-2 5

30

-3 5

-4 0

-4 5

4500 5000

Frequency,Hz

Figure 4.11 T argeting 2nd side lobe w ith truncation and w indow ing

4.4 Algorithm

Figure 4.12 shows the steps involved in the proposed algorithm. The algorithm is broken into four parts which are discussed next

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TRUNCATION

si (/) = s0(t)x red

SUPERIMPOSITION

NORMALISATION

S"(o)) = nJ\S'(ct))\ x exp{y'xZ5,'(<y)}

AVERAGING

DFT {Filter}

= D F T [D FT {S*(<y)} x DFT [Filter]^

F igure 4.12 S u perim p osition techn ique steps

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4.4.1 Truncation

The first step in the algorithm is obtaining the data sets from the original data. Depending on the side lobes being targeted, the truncation ratios are chosen using (4.10), (4.11) or (4.12). Equation (4.13) describes the truncation where the ith side lobe is being targeted.

st (0 = 5o(Ox re ^ j ~ _ ~ | (4.13)

Where r,. = kfa , ki is (4.10), (4.11) or (4.12) and s0 (?) is (4.5).

4.4.2 Superimposition

Equations (4.14) and (4.15) describe the second step in the algorithm. This step is where the spectra of the original data set and the derived data sets are superimposed. Equation (4.14) describes the Fourier transform of the data sets to be superimposed and (4.15) describes the superimposition which is a product of the spectra.

S( (a>) = exp( ) — A \ sine

Where A is the amplitude of the original signal, s0 (?).

s v ^ r r . . ^ )

Where n is the index for the spectra of the truncated data sets with n = 0 representing the spectrum of the original signal.

4.4.3 Normalisation

The third step is the normalisation step. The Fourier transforms in (4.14) introduces a scaling factor ri and the multiplication in (4.15) results in a A" factor and a

55

factor J'Jr, ex p(- j c o r j l ). Hence the normalisation step described by (4.16) is required

to /=o

maintain relative strengths between the frequency components in the signal and remove the complex factor in the spectrum. Note how S'{co) is separated into its magnitude and phase, this is important as the phase relationship between the truncated signals is what the averaging step described next depends on.

S\co) = ^\S'(co)\ x exp { j x ZS'(co)} x —exp f (4. 16)

,= 0 Ti V 2 J

4.4.4 Averaging

In the Fourier transform spectrum of a finite sinusoid successive side lobes in the spectrum will have a phase pattern of p + n, p, p + n, p, p + n and so on where p is a first order linear equation. The linear equation p comes from a complex factor introduced in the spectrum due to the finite and causal nature of the signal whereas the n part comes from the sine function alternating between sets of negative and positive value in successive lobes. For example the spectrum function (4.14) has a complex factor exp{-jco r^, if this factor is removed as is done in the normalisation step of the technique, (4.14) becomes (4.17), therefore when (4.17) is negative, its phase is n, when it is positive its phase is 0.

CO-Cl),

2 K

As the lobes alternate between negative and positive values as shown in original signal spectrum trace of Figure 4.13, the phase of the lobes will alternate between n and 0. The resulting side lobes from a targeted side lobe after superimposition will also follow a similar pattern as shown in the product of spectrums trace of Figure 4.13, where the and 2nd side lobes are targeted. Averaging over the width of 2 successive side lobes (i.e. a negative and a positive lobe) will therefore have a cancellation effect over the side lobes.

As the exact positions of the lobes are unknown the averaging is performed using a

56

uniform moving average filter [88]. The moving average filtering is a convolution described by

(4.18)

where G (co) = 1/^T0 red{cor0}, 1/r0 being the width of a side lobe in the original signal.

The actual implementation is done via a series of Fourier transforms using the convolution theorem [80] as described by (4.19)

S"(t») = F r _1[F r{S *(fi»)}xF r{G (© )}] (4.19)

Original Signal Spectrum Product of Spectrums

Moving Average Filter

I \ i

/

1 to, radians

Figure 4.13 A p p lying averagin g filter to prod uct o f spectrum s

So far the processing technique has been described for a single frequency signal. However, the processing technique described is linear, therefore using linear superimposition the technique can be applied to a multi-frequency signal.

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4.5 Discussion

In this section some general issues regarding the technique are discussed.