## Appendix C Development of the Time-Limited Fractional Frequency Shift  
In this appendix, the discrete frequency-domain characterization of the \"time-limited fractional frequency shift\" operation will be developed following the principles of its dual in discrete time-domain, namely the \"band-limited fractional time delay\" operation according to [19]. What makes it possible to synthesize a delay, \(\tau=pT\), which is only a fraction of the sample interval, \(T\), in a discrete-time system sampled at rate 1/\(T\), is the fact that this fractional delay needs to be effective only in a limited bandwidth around 0 Hz, at most as wide as the Nyquist band, \(B=1/T\). Therefore, from \(f=-1/(2T)\) to \(1/(2T)\), the frequency response \(H(f)\) should be all-pass, that is flat with phase factor \(e^{-j2\pi/\tau}\), and zero outside this band, making it low-pass. The corresponding impulse response in continuous-time is  
\[h(t)=\int\limits_{-\infty}^{\infty}H(f)e^{j2\pi ft}df=\int\limits_{-B/2}^{B/2 }e^{j2\pi f(t-\tau)}df \tag{10}\]  
which is straight-forward to evaluate as  
\[h(t)=B\frac{\sin(\pi B(t-\tau))}{\pi B(t-\tau)}\equiv BSince(B(t-\tau)) \tag{11}\]  
This defines the envelope of the discrete-time samples taken at integer multiples of \(T\) by the sampling impulses as  
\[h(nT)=\int\limits_{-\infty}^{\infty}h(t)\delta(t-nT)dt=BSinc(B(nT-\tau)) \tag{12}\]  
The envelopes and samples of the impulse response are shown below for both integer and fractional delays, computed according to Figure 3 of [19]. By substituting \(T=1/B\) and \(\tau=pT\), this becomes the discrete-time impulse response  
\[y(N)=B\sum_{n=0}^{N}x(N-n)Sinc(n-p) \tag{10}\]  
which can be implemented directly as a FIR filter whose coefficients correspond to integer values of \(n\) with \(0<p<1\) in the above summation. In practice the filter s truncated to a practical number of coefficients, \(N+1\), corresponding to \(N\) sample delays, and effects a total delay of \(N\)\(T/2+\tau\), because it must be causal. It is reproduced below according to Figure 4 of [19]. It can also be utilized in an adaptive multipath equalizer by adapting the coefficients according to a feedback algorithm that minimizes some statistical property of the signal error, as for example in [19]. A parallel development of the dual structure for the fractional frequency shift of \(\mu=q2F\) in frequency domain can now proceed as follows: denoting the time-gating function as \(g(t)=e^{j2\pi\mu t}\) for \(-A/2<t<A/2\) and 0 elsewhere, its spectrum in the continuous frequency-domain is the inverse Fourier transform  
\[G(f)=\int\limits_{-\infty}^{\infty}g(t)e^{-j2\pi f_{t}}dt=\int\limits_{-A/2}^{ A/2}e^{-j2\pi(f-\mu)t}dt \tag{10}\]  
which is seen to evaluate to  
\[G(f)=A\frac{\sin(\pi A(f-\mu))}{\pi A(t-\mu)}\equiv ASince(A(f-\mu)) \tag{11}\]  
This defines the envelope of the frequency-response of the fractional-shift operation, where it is clearly seen that its peak is shifted from the center 0 Hz to \(f=\mu\) Hz. When this gating window is repeated in time at intervals \(T/2\) (periodic extension in time), it results in spectral sampling at intervals \(2F=2/T\) expressed as  
\[ \begin{split} G(m2F)=\int\limits_{-\infty}^{\infty}G(f)\delta(f- m2F)df\cdots\\ \cdots=ASinc(A(m2F-\mu))\end{split}  \tag{12}\]  
finally resulting in the discrete-frequency response  
\[Y(M)=\frac{1}{2F}\sum\limits_{m=0}^{M}X(m-M)Sinc(m-q) \tag{13}\]  
with \(A=1/(2F)\) and \(\mu=q2F\). It is the dual of (C4) and defines the coefficients of the spectral replicas of the frequency-response of the time-gating function. A conceptual realization analogous to Figure 10 can be formulated for it, truncated to the \(M\)-th harmonic of \(2F\). Because the realization must be in (real-) time domain, the \"\(2F\)\" frequency-shift elements corresponding (in dual fashion) to the time-shift (sample delay) elements \(z^{-1}\) in Figure 10 are replaced by multipliers by complex factor \(e^{j2\pi 2 Ft}\), which effects the harmonics of 2 times the pseudo-Doppler modulating frequency, \(2F\).