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Figure 1. A typical sum-of-products operation.
N-1 __\ z =(1/N) * / x(n) * y(n)-- n = 0

A sum-of-products operation

This notation means that a new value for z is calculated by multiplying the first N corresponding samples from each of two numeric series (x(n) and y(n)),calculating the sum of the products, and dividing the sum by N.

(In this module, I will be dealing primarily with numeric series that represent samples from a continuous function taken over time. Therefore, Iwill often refer to the numeric series as a time series.)

An alternative notation

The above notation requires about six lines of text to construct, and therefore could easily become scrambled during the HTML publishing process. Ihave invented an alternative notation that means exactly the same thing, but is less likely to be damaged during the publishing process. My new notation isshown in Figure 2 . You should be able to compare this notation with Figure 1 and correlate the terms in the notation to the verbal description of the operationgiven above.

Figure 2. Alternative notation for a sum-of-products operation.
z = (1/N) * S(n=0,N-1)[(x(n) * y(n)]

This is the notation that I will use in this module.

Preview

What is a time series?

I discussed the concept of a time series in some detail in my module titled Dsp00104-Sampled Time Series . For purposes of this module, suffice it to say that a time series is a set of sample values taken from a continuous function at equal increments of timeover a specified time interval. For example, if you were to record the temperature in your office every minute for six hours, the set of 360 differentvalues that you would record could be considered as a time series.

A new time series is produced

In DSP, we often multiply two time series together on a sample by sample basis. When I multiply the values in the time series x(n) by thecorresponding values in the time series y(n), that produces a new time series, which I will call w(n).

Calculation of the mean or average

If I compute the sum of the individual values in the series w(n), and then divide that sum by the number of samples, this is nothing more than thecalculation of the mean or average value of the time series named w(n). Most DSP operations boil down to nothing more complicated than calculating the averagevalue of the product of two time series.

Knowing what to multiply, when, and why

The real trick in DSP is knowing what to multiply, when to multiply it, and why to multiply it.

Some DSP algorithms are very complex. For example, the Fast Fourier Transform (FFT) algorithm, involves nothing more than a lot of multiply-add operations under the control of an extremely complex and efficient control structure.

In this module, I will concentrate on the Discrete Fourier Transform (DFT) algorithm, which is much less complex and therefore much easier to understand.

(While the DFT and the FFT produce the same results, the DFT typically runs much more slowly than the FFT, which is optimized for speed.)

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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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