Recall Gabor Communication Theory and Joint Time-Frequency Analysis
Xiang-Gen Xia, University of Delaware
2025.8.31
Abstract
In this article, we first briefly recall Gabor’s communication theory and then Gabor transform and expansion, and also its connection with joint time-frequency analysis.
1. Gabor Communication Theory
In the 40s of the last century, there were two major concepts about digital communications. One is by Claude Shannon, i.e., the well-known Shannon communication theory published in the 1948 Bell System Tech Journal, which has become the foundation in modern digital communications. This is very well known these days and I will not elaborate any more about it here. The other is by Dennis Gabor who published “Theory of Communication” Part 1, Part 2, and Part 3 in 1946 [1]. In comparison, Shannon’s theory is more from the statistical point of view and more mathematical, while Gabor’s theory is more from structural or deterministic point of view and more physical, and also from more system view point. I will try to elaborate Gabor’s communication theory a little more below.
In Gabor’s communication theory, digital transmission is basically to tile the time and frequency plane. Given a time and frequency plane (or a channel), how many atoms (called Gabor atoms) can be packed and separated in the plane is kind of the physical capacity of the channel. The smaller the atoms are, the larger the capacity is, i.e., more different signals can be separated/detected. In other words, the smaller the product of the band width and time width is, the larger the capacity is. Unfortunately, due to Heisenberg’s uncertainty principle, the minimum product of the band width and the time width is ½, which in fact provides the physical capacity for a given time and frequency plane, no matter single or multi-user communication systems. This is the fundamental limit in physics and unfortunately, these days many people have forgotten this fundamental physics limit. I think that a key difference with Shannon communication theory is that in Gabor’s framework, the discrete signals Sn and Sn,k below are not necessarily quantized, i.e., not necessary in bits, and I think that it might be better to call Gabor’s as semi-digital.
Since Gaussian pulse p(t)= a exp(-bt2) for non-zero constants a and b>0 reaches the lower bound ½ of the product of the band width and time width, Gaussian pulse is the most compact pulse (or the most time and frequency localized pulse) in the product domain of time and frequency. This is the reason why it is used in the following transmission signal
(1)
where Sn are the digital symbols to transmit and T is the time duration of a digital symbol to transmit. In fact, it is the reason why Gaussian pulse is used in the 2G GSM standard. The transmission signal in (1) is only in time domain. If it is tiled in both time and frequency domains, the signal becomes
(2)
It is the formula (1.29) in [1], where p(t) is the Gaussian pulse. In other words, in terms of the time and frequency tilings, no pulse is better than the Gaussian pulse, no matter it is single or multi-user communication systems, for a given time and frequency/band rectangular window.
In fact, the frequency shift 1/T in (2) can be generalized to a general W as
(3)
under the condition 
, where the functions pn,k(t)=p(t-nT) exp(i2πkW) for integers n and k are called Gabor atoms. The transmission signal (2) is a special case of (3) when W=1/T. If we sample time variable t in (3) and use the rectangular pulse, i.e., p(t) is the rectangular pulse of length T, for example, t=l, T=N and W=1/N, (3) is an OFDM signal without cyclic prefix (CP). If p(t) is a more general pulse, (3) is generalized OFDM (GFDM). In other words, the form of GFDM may be traced back to Gabor's 1946 paper [1].
2. Short-Time Fourier Transform, Gabor Expansion, and Joint Time-Frequency Analysis
The two dimensional signal Sn,k in (3) can be thought of as a two dimensional sampling of the following short-time Fourier transform (STFT)
S(t,f )= ∫
(4)
where 
(t) is called an analysis window in time domain. STFT (4) is also called windowed Fourier transform. It was successfully applied in speech analysis in 1970’s. Then, STFT was extensively studied in the era of wavelets of the 1990’s as the first joint time-frequency analysis (JTFA) technique. Due to Gabor’s contribution, the two dimensional sampling Sn,k in (3) of the STFT S(t,f) of signal s(t) in (4):
∫ 
(5)
is called the Gabor transform of signal s(t) and the pulse p(t) in (3) is also called a synthesis window. The representation (3) is called Gabor representation (or expansion) of signal s(t). The pair (3) and (5) are called Gabor representation and transform pair. More on the sampling distances on times T and T’ and on frequences W and W’ for (3) and (5) to hold simultaneously was studied thoroughly by Daubechies in [2]. For finite length discrete time signals, it is called discrete Gabor transform (DGT) and inverse DGT (IDGT) [5]. For the relationship between the synthesis and analysis window functions p(t) and γ(t) in (3) and (5) either for analog time or discrete time, Wexler-Raz [3] obtained an identity called Wexler-Raz identity in analog time, discrete time, or finite length discrete time. The relationship corresponds to the biorthogonality, since the synthesis and the analysis windows p(t) and γ(t) may not be the same. In the oversampling case (in the finite length discrete time signal case, oversampling means that the number of points in the DGT domain is more than the signal length in time domain), for a given synthesis window p(t), the analysis window γ(t) satisfying the Wexler-Raz identity is not unique. The optimal γ(t) (called the most-orthogonal-like) was proposed in [4] and obtained in [5] for finite length discrete time signals, and in [6] for scaled most-orthogonal-like set-up in either discrete or continuous time that, interestingly, is the same as the one without scaling, i.e., the same as that obtained in [5]. Also, interestingly, the optimal most-orthogonal-like solution coincides with the minimum norm solution in either discrete time [5] or continuous time [6]. For a finite discrete time, the DGT of a finite length signal of 1 dimension becomes a 2 dimensional matrix. Its matrix rank was investigated and determined in [7].
A JTFA is for non-stationary signal analysis and usually concentrates a signal (such as a chirp signal) while spreads noise. A quantitative SNR analysis in a JTFA domain is given in [8,9,10]. As we know that one important purpose for signal transformation is to clean a noisy signal. When a signal is non-stationary, the traditional filtering in the Fourier domain may not work well, since the Fourier transform may not be able to concentrate the signal. In this case, one may apply a JTFA that may concentrate the signal while spreads the noise, and thus one may apply a filtering in the JTFA domain. However, since a JTFA is usually not an onto transform, after the filtering/clean-up in the JTFA domain, the filtered two dimensional signal may not correspond to a time domain signal, i.e., there may be no time domain signal whose DGT is the filtered one. Thus, filtering using JTFA (it is a time-variant filter) is not as trivial as that using the Fourier transform (it is a time-invariant filter). In this case, an iterative time-variant filter by turning back and forth between DGT and time domains using DGT/IDGT is proposed in [11] with the convergence analysis. It says that the convergence is good for the most-orthogonal-like window pairs. This DGT based time-variant filtering is used in system identification in low SNR environment in [12] by transmitting chirp signals.
No transform can be universally good for every signal, including JTFA. In my opinion, JTFA works well in particular for chirp type signals including high order chirps. A discrete chirp-Fourier transform (DCFT) is proposed in [13] for linear chirp signal matching, where it shows that DCFT works optimally when the signal length is a prime number. A good book on JTFA is [14] by Qian and Chen.
3. Conclusion
In this short article, we have briefly recalled Gabor communication theory in the 1940’s, which is from a different perspective of Shannon’s, and more from the physics and signal processing point of view, by packing signals in a time and frequency plane. In the later years, it has become Gabor atoms, Gabor transform and Gabor expansion/representation for nonstationary signal analysis as a major JTFA.
References
[1] D. Gabor, “Theory of communication: Part 1, Part 2, Part3,” Journal of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering, vol. 93, no. 26, Nov. 1946. https://jcsphysics.net/lit/gabor1946.pdf
[2] I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis," IEEE Transactions on Information Theory, vol. 36, no. 5, pp. 961-1005, Sept. 1990.
[3] J. Wexler and S. Raz. “Discrete Gabor expansions,” Signal Processing, vol. 21, no. 3, pp. 207-221, Nov. 1990.
[4] S. Qian, K. Chen, and S. Li, “Optimal biorthogonal sequence for finite discrete-time Gabor expansion,” Signal Processing, vol. 27, no. 2, pp. 177-185, May 1992.
[5] S. Qian and D. Chen, “Discrete Gabor transform,” IEEE Transactions Signal Processing, vol. 41, pp. 2429-2438, July 1993.
[6] X.-G. Xia, “On characterization of the optimal biorthogonal window functions for Gabor transforms,” IEEE Transactions on Signal Processing, vol. 44, no. 1, pp. 133-136, Jan. 1996.
[7] X.-G. Xia and S. Qian, “On the rank of the discrete Gabor transform matrix,” Signal Processing, vol.81, no. 5, pp. 1083-1087, May 2001.
[8] X.-G. Xia, “A quantitative analysis of SNR in the short-time Fourier transform domain for multicomponent signals,” IEEE Transactions on Signal Processing, vol.46, no. 1. pp. 200-203, Jan. 1998.
[9] X.-G. Xia and V. C. Chen, “A quantitative SNR analysis for the pseudo Wigner-Ville distribution,” IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2891-2894, Oct. 1999.
[10] X.-G. Xia, G. Wang, and V. C. Chen, “A quantitative signal-to-noise ratio analysis for ISAR imaging using joint time-frequency analysis --Short time Fourier transform,” IEEE Transactions on Aerospace and Electronics Systems, vol. 38, no. 2, pp. 649-659, April 2002.
[11] X.-G. Xia and S. Qian, “Convergence of an iterative time-variant filtering based on discrete Gabor transform,” IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2894-2899, Oct. 1999.
[12] X.-G. Xia, "System identification using chirp signals and time-variant filters in the joint time-frequency domain," IEEE Transactions on Signal Processing, vol. 45, no. 8, pp.2072-2084, Aug. 1997.
[13] X.-G. Xia, “Discrete chirp-Fourier transform and its application in chirp rate estimation,” IEEE Transactions on Signal Processing, vol. 48, no. 11, pp. 3122-3133, Nov. 2000.
[14] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, United Kingdom, PTR Prentice Hall, 1996.
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