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PDSP16510A MA

Arithmetic Accuracy

Max Tone

Slot Noise

2 Tones

WRT Noise

Test

with

Freq Spread

16 bit,unconditional

scaling

24 bit arithmetic with

unconditional scaling,

16 bit inputs

16 bit inputs with

PDSP16510 block FP

Full 32 bit Floating point

with 16 bit inputs

Table 8. Comparative Dynamic Range Measurements

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.

The ‘best’ dynamic range figures will be obtained with

single tone measurements, and these results are often quoted

to indicate the need for greater bit accuracies. The measure

is the ratio of a full scale sinusoid to the average noise level

and the results will be essentially independent of the window

operator. The results given by the PDSP16510 are compared

to various other configurations in the first column of Table 8.

With this method the dynamic range is bound to improve as

more bits are used to represent the data. Theoretically 6 dB of

dynamic range will be obtained for every bit representing the

input data, if the internal arithmetic accuracy gives no degra-

dation in performance. In practice this improvement has no

significance since the incoming waveforms will be much more

complex than a single sinusoid.

An alternative method of determining dynamic range is

with a slot noise test. White noise is passed through a narrow-

band notch filter, several frequency bins wide, and the FFT

computed. There is no noise in the filtered slot at the input to

the FFT, but there is noise in the frequency bins corresponding

to the width of the notch. Dynamic range is measured as the

difference in dB of the average signal power and the average

noise power and can be considered to give more useful

results. Comparative results from various configurations are

also given in the second column of Table 8. The performance

with 24 bit data is seen to be little better than that obtained with

the PDSP16510. This can be attributed to the scaling scheme,

word growth, and rounding method used within the device.

When two nearby tones are to be capable of detection,

the window operator will dictate the performance of the

system. The final column in Table 8 illustrates the results

obtained using two sinusoids of different amplitudes, with the

larger one residing mid-way between two frequency bins, and

the smaller 5.5 bins away. The two frequencies are five bins

apart to avoid the effects of the mainlobe widths. The dB

figures given are the difference in amplitude between the two

signals when the smaller one is still just detectable as a

separate peak from the larger one.

This technique illustrates the performance of the window,

since the amount by which sidelobe structure of the larger

signal swamps the mainlobe of the smaller signal will effect

whether the smaller signal is detected. The theoretical attenu-

ation of the highest sidelobe levels, with respect to the

mainlobe, for the window options provided by the PDSP16510

have been given in Table 7, and represent the dynamic range

that can be obtained if arithmetic effects are ignored. The

results in the final column in Table 8 are the practical results

given by the device, and as with the slot noise test indicate that

the arithmetic scheme used by the PDSP16510 is equivalent

to using 24 bit data. The Blackman Harris window was used

in all cases.

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 correla-

tion 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.

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 win-

dows 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. Resolu-

tion 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 deter-

mined 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 opera-

tor and the twiddle factors, and the way the growth in word

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