PDSP16515A

20

Examination of Table 7 shows that the Blackman-Harris

window gives performance very similar to that of the Kaiser-

Bessel and Dolph-Chebyshev windows. The latter two

windows can not be computed as they are needed since they

are mathematically too complicated. The values are normally

pre-computed and stored in a ROM; this would need to contain

1M bits to match the accuracy of the rest of the system.

Use of the Hamming window gives worse dynamic range than

the more complex windows, but it has less effect on the

overlap correlation and it has a smaller main lobe width.

Spectral Performance

There are two important parameters in the measurement of

spectral response: resolution and dynamic range. Resolution

defines how closely two sinusoids can be spaced in frequency

and still be identified; dynamic range defines how great the

difference in the amplitudes of the sinusoids may be and yet

the smaller one still identified. Resolution is determined by the

observation time [i.e. the width of the frequency bin] and the

window operator that is used. Dynamic range is also

determined by the window operator, but in a hardware

implementation it is also influenced by the number of bits used

to represent the data throughout the calculation.

The hardware effects include the accuracy of the A/D

converter, the number of bits representing the window

operator and the twiddle factors, and the way the growth in

word length is handled as the FFT calculation proceeds. The

obvious way to overcome these limitations is to use floating

point arithmetic; but in real life the accuracy of the A/D

converter is fixed and the sample size is limited. Floating point

arithmetic is thus an overkill solution for the majority of

applications. This is especially true for transform sizes up to

1024 points, which is the intended application area.

Figures given for the dynamic range of a system must be

carefully interpreted, since there is no exact definition of the

measurement. Three different ways of measuring dynamic

range have been investigated using 1024 point transforms.

Window

Highest

Mid-Point

Overall

6dB

Overlap Correlation

Operator

Side Lobe

Loss dB

Loss dB

Bandwidth

75%

50%

Rectangular

-13

3.92

3.92

1.21

75

50

Hamming

-43

1.78

3.1

1.81

70.7

23.5

Dolph-Chebyshev

-70

1.25

3.35

2.17

60.2

11.9

[C = 3.5]

Kaiser-Bessel

-69

1.02

3.55

2.39

53.9

7.4

[C = 3]

Blackman

-58

1.1

3.47

2.35

56.7

9

Blackman-Harris

-67

1.13

3.45

1.81

57.2

9.6

[3 term]

Table 7. Window Performance ( from The use of Windows for Harmonic Analysis. F J Harris. Proc IEEE Vol 66. Jan 1978 )

Overall loss

An overall figure for the reduction in signal to noise ratio

can be obtained by adding the mid-point loss to the reciprocal

of the equivalent noise power bandwidth in dB. It is a measure

of the ability of the window to detect single tones in broadband

noise. The variance between windows is less than 1dB.

6.0dB Bandwidth

This figure, expressed in bin widths, represents the ability of

the window to resolve two tones and should be as close to

unity as possible. As the highest sidelobe level is reduced, this

parameter tends to get worse, and a compromise must be

used when choosing a window.

Overlap Correlation

In many practical systems the squared magnitudes of

successive transforms are averaged to reduce the variance of

the measurements. If, however, a windowed FFT is applied to

non overlapping partitions of the sequence, data near the

boundaries will be ignored since the window exhibits small

values at those points. To avoid this loss partitions are usually

overlapped by 50% or 75%, which might, at first sight, remove

the need to average successive transforms. If non-windowed

transforms are overlapped by 75% or 50%, then 75% or 50%

of the data will be correlated. When windows are applied,

however, the data common to both transforms will be operated

upon by different portions of the window waveform. The

difference in these portions will dictate the amount of

correlation between overlapped data. At 50% overlap Table 7

shows that with all windows the data is virtually independent,

and successive averaging would still be needed. At 75%

overlap figures are obtained which are closer to the 75%

correlation obtained with no window.