Twindow = 600ms, which corresponds to frequency resolution dFwindow = 1/Twindow = 1.667 Hz. In order to get 1Hz resolution you should use window length of 1s.

Your sampling time corresponds to the largest frequency you can represent. While your time window represents the smallest frequency that can be shown.

I am still confused about this. Can I obtain frequency powers in 8 and 9Hz using the 600ms EEG data.

You'll obtain powers in 8.33Hz and 10Hz. You need to increase your window size.

No, you can't, the resolution is too limited to allow this, apart from the fact that 8 and 9Hz are not on the "grid" of your FFT frequencies. In fact the FFT calculates the output of a linear filtering operation. The main lobe of this filter is too wide to make the difference between 8Hz and 9Hz reliably. It is possible to calculate the energy centered around 9Hz, but this will also contain contributions of the energy around 8Hz and 10Hz.

You could increase your sampling rate rate to 2 sec since that would give you an FFT with resolution 1 Hz. Remember that the FFT is an even function. If you need more time resolution you could use an AR model.

When I am not mistaking, the resolution is limited by the duration in time, not by the method used to analyze it. As suich, I am not convinced that an AR model would offer an increased resolution without making some assumptions about the signal.

Frequency resolution is inversely proportional to the duration of the time window. This is valid for all techniques. AR spectral analysis does better than Fourier-based spectral analysis because it has other, more realistic assumptions about the behaviour of the signal outside the window: While Fourier assumes periodic behaviour AR assumes continuity of the average and a number of derivatives (the order of the model). Read Kay and Marple's famous paper on this...

Kay, SM and Marple Jr, SL (1981) Spectral Analysis -a modern perspective. Proc. IEEE, vol. 69, N.11, pp. 1380-1419.

On antileakage windows ,read ...

Harris, FJ On the use of windows for harmonic analysis with the discrete Fourier transform, Proc IEEE, 66, pp.51-83.

Best of luck!

You can get a more pleasing "curve" by interpolating the 1.6666 Hz samples in frequency. This may be "good enough" but only you will know that. It does not add information in an information theoretical point of view.

A simple way to do it is to pad the 600ms record of time samples with zero-valued samples out to 10 seconds total. Then compute the Discrete Fourier Transform, DFT, FFT, whatever to get samples in frequency spaced at 0.1Hz.

As others have pointed out, it's better to get more data if you can so that the time span of actual data is at least 1 second. But then too, you might ask: "what is so magic about an even-integer frequency value like 8, 9 or 10?"

As Fred Marshall rightly says above, zero padding gives you a smoother, 'more pleasing' curve' wit more points, but it DOESNOT improve spectral resolution (the ability to to separate or 'resolve' two closely spaced peaks). To 'resolve' well your 8 and 9 Hz you need a segment of the signal that is longer than 1 s. Resolution is approximately 1/NT where N is the number of sample and T is the sampling interval. with the length of the signal T0=NT=1s your resolution is borderline for 1 Hz. Use T0=4 or 8 seconds and you will resolve well your 8 and 9 Hz. Then zero pad at your heart's content to see more points on the plot of spectrum and confirm with your own eyes that zero padding just interpolates with a sinc weighting function what is already here but does not improve spectral resolution.

Best wishes.

Thank you all. It's clearer now.

Code and plot attached.

This discussion has come up in several forums with the only issue - I think - is an unwillingnes to define terms, and look at the result. Straight from Google:

Spectral resolution - Spectral resolution is the ability to resolve spectral features and bands into their separate components.

Further,

• Accuracy - designates how close a measured value is to the true quantity of what is being measured. 
• Resolution - is the smallest increment the system can display or measure. 

Attached is two sets of plots, each showwing the hamming windowed fft of the sum of two sinusiodal components, 1: 1 and 2 hz, 2: 1.125 and 2 hz. Sampled at 16sps, 64 samples.  The ffts of the same 64 samples are shown without zero padding, padded with 64 zeros and padded with 128 zeros.

1. Count - 4 intervals ("bins") separate the 1 and 2 Hz peak without padding, 8 and 16 separate the peaks as zero padding increases. This is added resolution.

2. The 1.125Hz component in the second plot has no resolvable peak, only a 0.25Hz plateau, without zero padding; adding zero padding clearly shows a defined peak at 1.125Hz, a clear improvement in acccuracy.

The FFT is an exercize in curve fitting, a Fourier expansion of the repeated waveform segment, uniquely determined by the finite segment.  Increasing the zero padding and the order of the FFT clearly provides component functions that better match the waveform components, and thus provide better resolution and accuracy.

 Bandwidth on the other hand is only related to sample rate.

I certainly agree that good definitions are important.  I'm eagerly awaiting hearing who was unwilling to go down that road. 

That said, I suppose that we all could use a lesson in better definitions when they're needed.  It's not heretical to say "not always".  We don't start every conversation with a dictionary for a reason.  It's called efficiency of communication.  And, to the casual listener, it might appear that we're being sloppy - which we are - but perhaps not in the context of the actual communication.

Let's take a look at this interesting example from James Walter Taylor:

In every case, the underlying sinusoids reach their peak spectral values on at least one of the frequency sample points.  So the example is a special case.  A more general case would be where the spectral peaks did not fall on a frequency sample point.  But, the example does nearly deal with this issue by using different zero-padding lengths such that two lengths both fail to hit the peak of the 1.125Hz sinusoid - and we can ignore the 3rd one that does.

If we're being careful with terminology then it's important to ask what one's purpose is. Such as:

- determine the peak frequency to some determined accuracy.

- determine the bandwidth or spectral shape to some determined accuracy.

- determine the magnitude at the peak frequency to some determined accuracy.

Linyling Yang didn't exactly say any of these things but, rather, an estimate of the PSD.  So, a function of frequency and, with the description, the desire for a particular frequency resolution was implied.  And, it's reasonable that we assume that the frequency magnitudes throughout be generated to some accuracy.  The latter is often harder to reach in the sense that there is always noise if one only specifies it relative to some idealized underlying signal.

Because this is often or at least sometimes hard to do, we are going to have to be satisfied with the results of some measurement process and devices - and perhaps to compare results.  So stating the "determined accuracy" with any assurance as implied above may be elusive.

Imagine: "I want the PSD of an EKG to within 0.1% at each and every frequency to a resolution of 0.1Hz."

In fact, the two accuracy measures are intertwined.  If the temporal window is short enough, spectral components will overlap in the analysis and their amplitudes will be perturbed accordingly.

Then there is the issue of how large a temporal window is too large?  This is a reasonalbe question to ask if the spectral contents are dynamic.  Maybe all you want is a snapshot.  In that case the spectral resolution is limited perforce.

As Taylor's plots clearly show, there is no improvement of frequency resolution as the zero-pad lengths are increased.  If there were then the frequency peaks would be narrower.  And, a common misperception that frequency sample rate and frequency resolution are the same thing is repeated.

A more interesting set of plots to demonstrate this point would have been created using 1Hz combined with sinusoids first at 1.0333Hz, 1.0666Hz, 1.1Hz, etc.

The ability to distinguish between even the presence of two sinuoids is determined by the length of the window and, the corresponding, spectral resolution.  Zero padding, as clearly shown, only adds interpolated points to the frequency samples but this is not properly referred to as spectral resolution.  A much better term is "frequency sampling rate" which gets us away from the resolution issue fairly well altogether.

(Imagine a famililiar and identical situation in the time domain.  We set of two firecrackers in sequence in an anechoic chamber.  The adjust the temporal spacing and repeat for a large number of individual such events.  We record the result through a bandpass amplifier and, for convenience, sample, quantize and store the results on a computer in an .mpg file.  If the two explosions occur in a time span that is much less than the reciprocal of the amplifier bandwidth, we will find we have recorded a single implusive sound.  As the time between the explosions increases, eventually we can detect (either by ear or oscilloscope) that there are two distinct events.    The times where we can and cannot distinguish are determined by the amplifier bandwidth and NOT by the temporal sample rate - well, assuming that the sampling at least meets the Nyquist sampling criterion.  Raising the temporal sampling rate has NO effect on our ability to distinguish.  In the same sense, increasing the frequency sample rate has no effect on our ability to distinguish the presence of two sinusoids close in frequency.)

Whenever bullshit flows like this, one needs to examine the terms of the discussion.  In blunt terms -

Resolution - is the smallest increment the system can display or measure. Spectral resolution is points or bins per Hertz, the precision of a measurement of frequency.  A well defined engineering term.

Accuracy - designates how close a measured value is to the true quantity of what is being measured.

We might mean the precision of the determination of frequency, but this is almost identical to resolution, more usually spectral accuracy is the relative precision of magnitude of a frequency component. Another well defined engineering term, but one which is slightly less applicable to this discussion, since any FFT is a precise, invertible, 1:1 mapping from the set of finite, real signal vectors to the complex plane.  Here we're really talking about a comparison of the significant components of the FFT to the continuous Fourier transform of the entire, continuous signal, I suspect.

Increasing the FFT order, whether by zero padding or increasing the length or the waveform interval, at the same sample rate, always increases *both*.  You provide a richer set of basis functions in a higher order FFT and your resolution and accuracy improve.  Shown clearly by the example.  

On the other hand, the FFT is not spectral estimation in any intuitive, clear, or optimal sense.  Also demonstrated by the example, since increasing the order of the FFT in no way causes the spectra to approach the expected delta function spectral form.