## Why an Impulse is Not White Noise

Given the test for white noise that its sample autocorrelation must
approach an impulse in the limit, one might suppose that the
*impulse* signal
is technically ``white noise'',
because its sample autocorrelation function is a perfect impulse.
However, the impulse signal fails the test of being
*stationary*. That is, its statistics are not the same at every
time instant. Instead, we classify an impulse as a
*deterministic* signal. What is true is that the impulse
signal is the deterministic counterpart of white noise. Both signals
contain all frequencies in equal amounts. We will see that all
approaches to noise spectrum analysis (that we will consider)
effectively replace noise by its autocorrelation function, thereby
converting it to deterministic form. The impulse signal is already
deterministic.

We can modify our white-noise test to exclude obviously nonstationary
signals by dividing the signal under analysis into
blocks and
computing the sample autocorrelation in each block. The final sample
autocorrelation is defined as the average of the block sample
autocorrelations. However, we can also test to see that the blocks
are sufficiently ``comparable''. A precise definition of
``comparable'' will need to wait, but intuitively, we expect that the
larger the block size (the more averaging of lagged products within
each block), the more nearly identical the results for each block
should be. For the impulse signal, the first block gives an ideal
impulse for the sample autocorrelation, while all other blocks give
the zero signal. The impulse will therefore be declared
*nonstationary* under any reasonable definition of what it means
to be ``comparable'' from block to block.

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The Periodogram

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