# literature related to Spectral resolution

I am interested in spectral resolution of spectral estimation methods.

Can you point me to some good literature, videos or books from which i can read and study?

I want to understand how to resolve two closed spaced sinusoids: $$x_1(t) = \sin(\omega_1 t + \phi_1) \\ x_2(t) = \sin(\omega_2 t + \phi_2) \\$$ where $\omega_1 \approx \omega_2$.

• Your question is very vague, and cannot be easily answered. Please define what you mean by resolution capacity and what specifically you need to know. – Peter K. Mar 25 '14 at 13:45
• i want methods and generally theoretical approaches for spectral resolution,by spectral resolution i mean when we have closed spaced deterministic components according to their frequency and amplitudes,what methods exist for their differentiation or how to distinguish from each other this components – dato datuashvili Mar 25 '14 at 13:52
• Please update your question, then you will be able to select it for re-opening. "Resolution capacity" didn't make much sense to me. "Spectral resolution" makes much more sense. – Peter K. Mar 25 '14 at 15:01
• i have changed ,could you see please – dato datuashvili Mar 25 '14 at 15:55
• yes even instead of two sinusoid,it could be a lot of – dato datuashvili Mar 25 '14 at 18:31

I would recommend
Array Signal Processing - Johnson, Dudgeon. This book covers classical spectral estimation, Minimum Variance Distortionless Response (MVDR), Linear Prediction, and subspace methods (e.g. MUSIC and ESPIRIT). It provides examples of the resolution capabilities between these methods. The are quite a number of good references in this book if you require additional detail.

MUSIC and ESPIRIT type algorithms are dicussed in most advanced DSP texts:
Discrete Random Signals and Statistical Signal Processing, Therrien

For parametric methods:
Spectral Analysis - A modern Perspective (Kay, Marple) Proc. IEEE Vol 6, No 11, 1981
Modern Spectral Estimation: Theory and Application, Steven Kay
Digital Spectral Analysis - S.L. Marple
SPECTRAL ANALYSIS OF SIGNALS - P Stoica and R Moses PDF is available here

The books by Marple and Kay tend to focus on algorithms i.e. implementations of various MA, AR, and ARMA approaches (e.g. fast Lattice filter implementations) rather than the performance of a basic approach

Compressive Sensing / Sparse Reconstruction techniques can also be used. There are too many references here to list. There are a few books by these authors Michael Elad, Yonina Eldar, and Holger Rauhut. These techniques are often quoted as "having super resolution properties"

The algorithms are often hard to compare directly because they all have twiddle factors which affect their performance. For example
1. For MA, AR, ARMA, what order of model to use?
2. For Compressive techniques - how closely to space the template vectors? What level of regularization to use? How many frequencies are there? How many samples do we have?

• but also AR,MA,ARMA model is used only for stationary model right? – dato datuashvili Mar 27 '14 at 18:47

You could start by looking into the MUSIC and ESPIRIT algorithms for constrained parameter estimation.

Look into the "Chirp z transform" (somtimes referred to as the zoom FFT"). You can go to a good standard DSP reference. I believe you can find this as public domain on the internet. The book is:

"Digital Signal Processing" by Oppenheim and Schafer See chapter 6, section 6.

This techniques allows you to concentrate you're calculations on a narrow range of frequencies. You would "zoom" into the range of your two closely spaced sine waves.

You can also google "chirp z transform", "zoom z transform" or "zoom FFT".

• is this book free on internet?if you can give me links please – dato datuashvili Mar 26 '14 at 4:37
• i have one question if i have signal,what i should know in advance so that usage of Chirp z transform was effective? – dato datuashvili Mar 26 '14 at 5:16
• Just for the record: the Chirp Z-transform and the zoom FFT are not the same thing! – Matt L. Mar 26 '14 at 21:22
• I've seen the two terms used interchangeably, though this may be poor usage. Can you direct me to a technical presentation of the zoom z transform. The discussion in the reference I provided is on the chirp z transform. – user2718 Mar 27 '14 at 1:28
• www-elec.inaoep.mx/~rogerio/… See chaper 9. This is the same author and contains much of the same material. – user2718 Mar 27 '14 at 1:33