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(Note however, that you may not see the zero-valued points in the spectrum if you don't compute the spectral values at exactly those frequencyvalues. This is the case for some of the plots in Figure 11 .)

The frequency resolution of the Fourier transform

Some analysts regard a frequency interval equal to the reciprocal of the length of the time series as being the useful resolution of the spectrumanalysis process.

In other words, two peaks in the spectrum cannot be resolved if the frequency difference between them is less than the reciprocal of the length of the timeseries.

Illustration of frequency resolution

This is illustrated by the plots in Figure 15 . Figure 15 is similar to Figure 13 with one major difference. In Figure 13 , the frequency difference between the two sinusoids that made up each of the time series was rather large. In Figure 15 , the frequency difference between the two sinusoids that made up each of the time series was reduced to 1/400, (the reciprocal of the length of the longest time series).

Figure 15. Illustration of frequency resolution.
missing image

Each plot in Figure 15 shows the first 50 points produced by performing a Fourier transform on one of the time series. In each case, the time seriesconsisted of the sum of two sinusoids with a frequency separation of 1/400.

A match for the frequency resolution

The length of the time series for the bottom plot was 400 samples. Thus, the separation of the two sinusoids matched the frequency resolution available by performing a Fourier transform on that time series.

As you can see, the two peaks in the spectrum were resolved by the bottom plot in Figure 15 .

Insufficient frequency resolution

The other four time series were shorter, having lengths of 80, 160, 240, and 320 samples respectively, from top to bottom.

The important thing to note in Figure 15 is that the spectrum analysis performed on the 400-sample time series was successful in separating the twopeaks.

However, even though the spectrum analysis on the 320-sample time series hinted at a separation of the peaks, none of the spectrum analyses on the timeseries that were shorter than 400 samples successfully separated the peaks.

This illustrates that the frequency resolution of the Fourier transform is the reciprocal of the length of the time series.

Sufficient resolution in all five cases

I'm going to show you one more picture and then call it a wrap for this module. Figure 16 is similar to Figure 15 with one major difference.

Figure 16. Illustration of frequency resolution.
missing image

As before, the five plots in Figure 16 show the first 50 points produced by performing a Fourier transform on five different time series. Starting at thetop, the lengths of the time series were 80, 160, 240, 320, and 400 samples.

Adequate frequency resolution in all cases

Also as before, each time series was the sum of two sinusoids with closely-spaced frequencies. However, in Figure 16 , the difference between the sinusoidal frequencies was different from one time series to the next.

In Figure 16 , the frequency difference for the sinusoids contained in each time series was the reciprocal of the length of that particular time series.Therefore, the frequency difference for each case matched the frequency resolution of the Fourier transform.

The frequency of the lower-frequency peak was the same in all five cases. Therefore, this peak should line up vertically for the five plots in Figure 16 .

The frequency difference between the sinusoids was achieved by increasing the higher frequency by an amount equal to the reciprocal of the length of the timeseries.

Peaks were resolved in all five cases

If you examine Figure 16 , you will see that the peaks corresponding to the two sinusoids were resolved for all five time series.

As would be expected, the peaks appear to be broader for the shorter time series having the lower frequency resolution. The peaks are also separated inall five cases. However, the peaks for the lower-frequency sinusoid don't exactly line up vertically. Thus we see a small amount of measurement error inthe positions of the peaks


This module has presented a pseudo-mathematical discussion of issues involving the averaging of time series, and the impact of those issues onspectrum analysis.

Those averaging issues have an impact on many other areas of DSP as well, but the detrimental effect is probably more obvious in spectrum analysis than in otherareas.


This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: Dsp00108: Digital Signal Processing (DSP) in Java, Averaging Time Series
  • File: Dsp00108.htm
  • Published: 12/11/02

Baldwin begins with a discussion of averaging time series, and ends with a discussion of spectral resolution, covering several related topics in between.


Financial : Although the Connexions site makes it possible for you to download a PDF file for thismodule at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should beaware that some of the HTML elements in this module may not translate well into PDF.

I also want you to know that, I receive no financial compensation from the Connexions website even if you purchase the PDF version of the module.

In the past, unknown individuals have copied my modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.com showing me as the author. Ineither receive compensation for those sales nor do I know who does receive compensation. If you purchase such a book, please beaware that it is a copy of a module that is freely available on cnx.org and that it was made and published withoutmy prior knowledge.

Affiliation : I am a professor of Computer Information Technology at Austin Community College in Austin, TX.


Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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