CN1574805A - Frequency shift estimator for multicarrier receiver and method thereof - Google Patents

Abstract

The invention discloses a frequency deflection estimator of the multi-carrier receiver and the frequency deflection estimating method; wherein, the frequency deflection estimator comprises: a first signal generator, a delayer, a second signal generator, a multiplier and a deflection detector; wherein, the first signal generator makes use of I component and Q component of the synchronous signals to produce a complex signal; the delayer is used to delay the preset time for the produced complex signal and outputs the delayed complex signal; the second signal generator produces a conjugated complex signal corresponding to the delayed complex signal; the multiplier uses the complex signal and the conjugated complex signal and stabilizes the phase component of the complex signal; the deflection detector detects the frequency deflection on the basis of the phase component of a constant complex signal. Therefore, the frequency deflection estimator has the advantages that no matter whether the synchronous signal is detected accurately or not, the frequency deflection can be estimated.

Description

The frequency offset estimator and the method thereof that are used for multi-carrier receiver

Technical field

The present invention relates to a kind of frequency offset estimator of multi-carrier receiver, and be particularly related to a kind of frequency offset estimator and method of estimation thereof that the synchronizing signal in territory is in use carried out estimated frequency skew in the synchronous multi-carrier receiver that be used for.

Background technology

Tradition OFDM (OFDM) system has a kind of structure; wherein; the data arrangement of launching with frequency domain in the corresponding position of each subcarrier on; IFFT is modulated to ofdm signal on the time domain with the ofdm signal on the frequency domain; and will protect at interval (GI) to be inserted into the front of the ofdm signal after the modulation in the time domain, so that minimize intersymbol interference.

Simultaneously, (the time domain synchronization of TDS in ofdm signal, Domain Synchronous)-signal structure of OFDM is illustrated among Fig. 1, wherein GI is inserted into the front of the ofdm signal after the modulation on the time domain, and pseudo noise sequence (being referred to as " PN sequence " hereinafter), be Sync, be inserted into the front of GI.

That is, in the TDS-OFDM receiver, TDS-OFDM signal as shown in Figure 1 is launched, and by using the PN sequence on the time domain to carry out synchronously.The PN sequence is the synchronizing signal in the TDS-OFDM signal that is inserted on the time domain.

It is that the PN sequence that is included in the synchronizing signal in the TDS-OFDM signal has the characteristic that changes with each OFDM symbol.Thereby, for the estimated frequency skew, just need obtain the PN sequence exactly.

Therefore, for estimated frequency skew exactly in traditional TDS-OFDM receiver, just need obtain the PN sequence that changes with each OFDM symbol exactly.

Summary of the invention

One aspect of the present invention is to provide a kind of accuracy of not considering resulting synchronizing signal and the frequency offset estimator and the frequency offset estimation methods thereof of estimated frequency skew.

In order to realize above-mentioned aspect, comprise according to the frequency offset estimator of multi-carrier receiver of the present invention: first signal generator produces complex signal by I component and the Q component that uses synchronizing signal; Delayer, the complex signal scheduled time that delay is produced, the complex signal of output delay then; The secondary signal generator produces and the corresponding conjugate complex signal of complex signal that postpones; Multiplier uses complex signal and conjugate complex signal to make that the phase component of complex signal is constant; And offset detector, be the phase component of the complex signal of constant based on it, detect frequency shift (FS).

The complex signal S that in first signal generator, produces ₁(t) obtain by following formula.

$S_1\left(t\right)=({I^2}_{\mathrm{PN}}\left(t\right)-{Q^2}_{\mathrm{PN}}\left(t\right))+j2(I_{\mathrm{PN}}\left(t\right)\×Q_{\mathrm{PN}}\left(t\right))$

$=B\left(t\right)\{\cos \left(2{\mathrm{\Δ\ω}}_{c}t\right)+j\sin \left(2\mathrm{\Δ}{\mathrm{\ω}}_{c}t\right)\}$

$=B\left(t\right)\·e^{j2\mathrm{\Δ}{\mathrm{\ω}}_{c}t}$

Wherein, B (t) be by

$\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

Obtain, and PN (n) expression synchronizing signal, g (t-nT) expression is by the pulse signal of pulse shaping filter shaping.

The complex signal S that has postponed scheduled time D by delayer ₂(t) obtain by following formula.

$S_2\left(t\right)=B(t-D)e^{j(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t-2\mathrm{\Δ}{\mathrm{\ω}}_{C}D)}$

The conjugate complex signal S that in the secondary signal generator, produces ₂ ^*(t) obtain by following formula.

${S_2}^{*}\left(t\right)=B(t-D)e^{-j(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t-2\mathrm{\Δ}{\mathrm{\ω}}_{C}D)}$

Phase component according to the complex signal of multiplier output makes it become the signal S of constant ₃(t) obtain by following formula, and offset detector is from phase component 2 Δ ω _CDetect Δ ω among the D _C

$S_3\left(t\right)=S_1\left(t\right)\·{S_2}^{*}\left(t\right)=B^2\left(t\right)\·e^{j2\mathrm{\Δ}{\mathrm{\ω}}_{C}D}$

Frequency offset estimation methods according to multi-carrier receiver of the present invention comprises: first signal produces step, produces complex signal by I component and the Q component that uses synchronizing signal; Postpone step, the complex signal scheduled time that delay is produced, output then; Secondary signal produces step, produces the conjugate complex signal corresponding to the complex signal that postpones; Utilize complex signal and conjugate complex signal to make the constant step of phase component of complex signal; And the detection step, be that the phase component of the complex signal of constant detects frequency shift (FS) based on it.

Produce the complex signal S that produces in the step at first signal ₁(t) obtain by following formula.

$S_1\left(t\right)=({I^2}_{\mathrm{PN}}\left(t\right)-{Q^2}_{\mathrm{PN}}\left(t\right))+j2(I_{\mathrm{PN}}\left(t\right)\×Q_{\mathrm{PN}}\left(t\right))$

$=B\left(t\right)\{\cos \left(2{\mathrm{\Δ\ω}}_{c}t\right)+j\sin \left(2\mathrm{\Δ}{\mathrm{\ω}}_{c}t\right)\}$

$=B\left(t\right)\·e^{j2\mathrm{\Δ}{\mathrm{\ω}}_{C}t}$

Wherein, B (t) be by

$\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

Obtain, and PN (n) expression synchronizing signal, g (t-nT) expression is by the pulse signal of pulse shaping filter shaping.

The complex signal S that in postponing step, has postponed the scheduled time ₂(t) obtain by following formula.

$S_2\left(t\right)=B(t-D)e^{j(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t-2\mathrm{\Δ}{\mathrm{\ω}}_{C}D)}$

Produce the conjugate complex signal S that produces in the step in secondary signal ₂ ^*(t) obtain by following formula.

${S_2}^{*}\left(t\right)=B(t-D)e^{-j(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t-2\mathrm{\Δ}{\mathrm{\ω}}_{C}D)}$

Phase component according to the complex signal of multiplier output makes it become the signal S of constant ₃(t) obtain by following formula, and detect step from phase component 2 Δ ω _CDetect Δ ω among the D _C

$S_3\left(t\right)=S_1\left(t\right)\·{S_2}^{*}\left(t\right)=B^2\left(t\right)\·e^{j2\mathrm{\Δ}{\mathrm{\ω}}_{C}D}$

Therefore, use, need not search for PN sequence accurately, just can estimate to be offset with compensating frequency according to frequency offset estimator of the present invention.

Description of drawings

After having read following detailed in conjunction with the accompanying drawings, above aspect of the present invention, characteristic and advantage will become clearer, wherein:

Fig. 1 shows the structure of traditional TDS-OFDM signal;

Fig. 2 shows the block diagram according to frequency offset estimator of the present invention;

Fig. 3 shows the flow chart according to frequency offset estimation methods of the present invention.

Embodiment

Below, by the reference accompanying drawing, it is clear that the present invention will become.

Fig. 2 is the schematic block diagram according to frequency offset estimator of the present invention.With reference to Fig. 2, described according to frequency offset estimation algorithm of the present invention.

Frequency offset estimator 510 comprises: first signal generator 511; Delayer 513; Secondary signal generator 515; Multiplier 517; And offset detector 519.

First signal generator 511 produces complex signal by I component and the Q component that uses the PN sequence, and above-mentioned PN sequence is the synchronizing signal of input.That is, the complex signal that is produced has real part, its be I component the quadratic sum Q component square poor; And imaginary part, it is the poor of I component and Q component.

The complex signal scheduled time D that delayer 513 postpones generation in first signal generator 511, the signal of output delay then.That is because the complex signal that produces in first signal generator 511 is the trigonometric function of time, with its delay scheduled time so that the constant term of relevant skew and time have nothing to do.

Secondary signal generator 515 is created in the conjugate complex signal of the complex signal that postpones in the delayer 513.

Complex signal that multiplier 517 will produce in first signal generator 511 and the conjugate complex signal multiplication that in secondary signal generator 515, produces.That is, with complex signal and the conjugate complex signal multiplication with this complex signal of constant term, the signal of output has the constant term of relevant frequency shift (FS).

Offset detector 519 detects frequency shift (FS) based on the constant term of the relevant frequency shift (FS) of exporting from multiplier 517.

Thereby, use estimated frequency shift that frequency shift (FS) is compensated.

Fig. 3 is the flow chart according to frequency offset estimation methods of the present invention.With reference to following formula, frequency offset estimation methods has been described.

It is the I component and the Q component generation complex signal of the PN sequence of synchronizing signal that first signal generator 511 uses it.That is, produce complex signal S ₁(t), it has real part, its be I component the quadratic sum Q component square poor; And imaginary part, it is poor (S310) of I component and Q component.

For example, if frequency shift (FS) Δ ω is arranged in the PN sequence _CI (1) composition and Q (2) composition are obtained by following formula.

[formula 1]

$I_{\mathrm{PN}}\left(t\right)=\cos \left(\mathrm{\Δ}{\mathrm{\ω}}_{C}t\right)\underset{n}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)g(t-\mathrm{nT})\·\·\·\·\·\·\left(1\right)$

$Q_{\mathrm{PN}}\left(t\right)=\sin \left(\mathrm{\Δ}{\mathrm{\ω}}_{C}t\right)\underset{n}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)g(t-\mathrm{nT})\·\·\·\·\·\·\left(2\right)$

Wherein, PN (n) represents synchronizing signal, and g (t-nT) expression is by the pulse signal of pulse shaping filter shaping.

The complex signal S that in first signal generator 511, produces ₁(t) real part be the I component that calculates by formula 2 to 4 the quadratic sum Q component square between difference obtain.

[formula 2]

I ² _PN(t)

$=\mathrm{co}{s}^2\left({\mathrm{\Δ\ω}}_{C}t\right)\underset{n}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)g(t-\mathrm{nT})\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(m\right)g(t-\mathrm{mT})$

$={\cos }^2\left({\mathrm{\Δ\ω}}_{C}t\right)\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})$

$=\frac{1}{2}(1+\cos \left(2{\mathrm{\Δ\ω}}_{C}t\right))\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

[formula 3]

${Q^2}_{\mathrm{PN}}\left(t\right)$

$={\sin }^2\left({\mathrm{\Δ\ω}}_{C}t\right)\underset{n}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)g(t-\mathrm{nT})\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(m\right)g(t-\mathrm{mT})$

$={\sin }^2\left({\mathrm{\Δ\ω}}_{C}t\right)\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})$

$=\frac{1}{2}(1-\cos \left(2{\mathrm{\Δ\ω}}_{C}t\right))\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

[formula 4]

${I^2}_{\mathrm{PN}}\left(t\right)-{Q^2}_{\mathrm{PN}}\left(t\right)$

$=\cos \left(2{\mathrm{\Δ\ω}}_{C}t\right)\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

Simultaneously, the imaginary part of the complex signal that produces in first signal generator 511 is to be obtained by following formula 5.

[formula 5]

$I_{\mathrm{PN}}\left(t\right)\×Q_{\mathrm{PN}}\left(t\right)$

$=\sin \left(\mathrm{\Δ}{\mathrm{\ω}}_{C}t\right)\·\cos \left(\mathrm{\Δ}{\mathrm{\ω}}_{C}t\right)\left\{\underset{n}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)g(t-\mathrm{nT})\mathrm{PN}\left(m\right)g(t-\mathrm{mT})\right\}$

$=\frac{1}{2}\sin \left(2{\mathrm{\Δ\ω}}_{C}t\right)\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

With reference to formula 4 and 5, the complex signal of generation can be reduced to following formula 6.

[formula 6]

Suppose,

$B\left(t\right)=\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$ Then:

$S_1\left(t\right)=({I^2}_{\mathrm{PN}}\left(t\right)-{Q^2}_{\mathrm{PN}}\left(t\right))+j2(I_{\mathrm{PN}}\left(t\right)\×Q_{\mathrm{PN}}\left(t\right))$

$=B\left(t\right)\{\cos \left(2{\mathrm{\Δ\ω}}_{C}t\right)+j\sin \left(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t\right)\}$

$=B\left(t\right)\·e^{j2{\mathrm{\Δ\ω}}_{C}t}$

That is the complex signal S that in first signal generator 511, produces, ₁(t) obtain by formula 6, and be input to delayer 513.

Delayer 513 outputs have been delayed the complex signal S of scheduled time D ₂(t) (S330).The signal that postpones obtains by following formula 7.

[formula 7]

$S_2\left(t\right)=B(t-D)e^{j(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t-2\mathrm{\Δ}{\mathrm{\ω}}_{C}D)}$

Secondary signal generator 515 produce with from the delay of delayer 513 outputs the corresponding conjugate complex signal of complex signal S ₂ ^*(t) (S350).

The complex signal S that multiplier 517 will produce in first signal generator 511 ₁(t) the conjugate complex signal S that produces and in secondary signal generator 515 ₂ ^*(t) multiply each other, and make complex signal S ₁(t) phase component becomes and irrelevant constant term (S370) of time.The output signal S of multiplier 517 ₃(t) be to obtain by following formula 8.

[formula 8]

$S_3\left(t\right)=B\left(t\right)\·e^{j2\mathrm{\Δ}{\mathrm{\ω}}_{C}t}\·B(t-D)\·e^{-j(2\mathrm{\Δ}{\mathrm{\ω}}_{C}t-2\mathrm{\Δ}{\mathrm{\ω}}_{C}D)}=B^2\left(t\right)\·e^{j2\mathrm{\Δ}{\mathrm{\ω}}_{C}D}$

Offset detector 519 uses from the signal S in the formula 8 ₃(t) phase component 2 Δ ω _CD, and the frequency shift (FS) Δ ω of the PN sequence of detection input _C(S390).

According to said frequencies skew algorithm for estimating, needn't search for PN sequence accurately, just estimated frequency skew exactly, thus compensate this frequency shift (FS).

According to the present invention and since do not consider its be synchronizing signal the PN sequence accurate detection and this frequency shift (FS) is estimated, so this frequency shift (FS) provides several effects.

A kind of effect according to the present invention is that no longer needs are searched for PN sequence accurately.Another effect is that Frequency offset estimation is irrelevant with the PN sequence that changes with each symbol.Another effect is the error that has prevented the Frequency offset estimation that caused by the inaccurate detection of PN sequence.Another effect is that the circuit that does not need to separate is realized so that search for the PN sequence that changes with each symbol more accurately.

Though described the preferred embodiments of the present invention, in case those skilled in the art has acquired basic invention principle, other variation and modification can take place in this embodiment.Therefore, claims should be interpreted as comprising this most preferred embodiment and whole other variations and modification in spirit and scope of the invention.

Claims (10)

1. the frequency offset estimator of a multi-carrier receiver comprises:

First signal generator produces complex signal by I component and the Q component that uses synchronizing signal;

Delayer with the complex signal delay scheduled time that is produced, and is exported the complex signal of this delay;

The secondary signal generator produces the corresponding conjugate complex signal of complex signal with this delay;

Multiplier by using this complex signal and this conjugate complex signal, makes the phase component of this complex signal constant; And

Offset detector is that the phase component of the complex signal of constant detects frequency shift (FS) based on it.

2. frequency offset estimator as claimed in claim 1, wherein, the complex signal S that in first signal generator, produces ₁(t) obtain by following formula:

$S_1\left(t\right)=({I^2}_{\mathrm{PN}}\left(t\right)-{Q^2}_{\mathrm{PN}}\left(t\right))+j2(I_{\mathrm{PN}}\left(t\right)\×Q_{\mathrm{PN}}\left(t\right))$

$=B\left(t\right)\{\cos \left(2{\mathrm{\Δ\ω}}_{c}t\right)+j\sin \left(2{\mathrm{\Δ\ω}}_{c}t\right)\}$

$=B\left(t\right)\·e^{j2{\mathrm{\Δ\ω}}_{C}t}$

Wherein, B (t) be by

$\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

Obtain, and PN (n) expression synchronizing signal, g (t-nT) expression is by the pulse signal of pulse shaping filter shaping.

3. frequency offset estimator as claimed in claim 1 wherein, has been postponed the complex signal S of scheduled time D by delayer ₂(t) obtain by following formula:

$S_2\left(t\right)=B(t-D)e^{j(2{\mathrm{\Δ\ω}}_{C}t-2{\mathrm{\Δ\ω}}_{C}D)}$

4. frequency offset estimator as claimed in claim 1, wherein, the conjugate complex signal S that in the secondary signal generator, produces ₂ ^*(t) obtain by following formula:

${S_2}^{*}\left(t\right)=B(t-D)e^{-j(2{\mathrm{\Δ\ω}}_{C}t-2{\mathrm{\Δ\ω}}_{C}D)}$

5. frequency offset estimator as claimed in claim 1 wherein, makes it become the signal S of constant according to the phase component of the complex signal of multiplier output ₃(t) obtain by following formula, and this offset detector is from phase component 2 Δ ω _CDetect Δ ω among the D _C:

$S_3\left(t\right)=S_1\left(t\right)\·S_2*\left(t\right)=B^2\left(t\right)\·e^{j2{\mathrm{\Δ\ω}}_{C}D}$

6. the frequency offset estimation methods of a multi-carrier receiver comprises:

First signal produces step, produces complex signal by I component and the Q component that uses synchronizing signal;

Postpone step, with the complex signal delay scheduled time that is produced, output then;

Secondary signal produces step, produces the corresponding conjugate complex signal of complex signal with this delay;

By the constant step of phase component of utilizing this complex signal and this conjugate complex signal to make this complex signal; And

Detecting step, is that the phase component of this complex signal of constant detects frequency shift (FS) based on it.

7. frequency offset estimation methods as claimed in claim 6 wherein produces the complex signal S that produces in the step at first signal ₁(t) obtain by following formula:

$S_1\left(t\right)=({I^2}_{\mathrm{PN}}\left(t\right)-{Q^2}_{\mathrm{PN}}\left(t\right))+j2(I_{\mathrm{PN}}\left(t\right)\×Q_{\mathrm{PN}}\left(t\right))$

$=B\left(t\right)\{\cos \left(2{\mathrm{\Δ\ω}}_{c}t\right)+j\sin \left(2{\mathrm{\Δ\ω}}_{c}t\right)\}$

$=B\left(t\right)\·e^{j2{\mathrm{\Δ\ω}}_{C}t}$

Wherein, B (t) be by

$\{\underset{n}{\mathrm{\Σ}}{\mathrm{PN}}^2\left(n\right)g^2(t-\mathrm{nT})+\underset{n}{\mathrm{\Σ}}\underset{m}{\mathrm{\Σ}}\mathrm{PN}\left(n\right)\mathrm{PN}\left(m\right)g(t-\mathrm{nT})g(t-\mathrm{mT})\}$

Obtain, and PN (n) expression synchronizing signal, g (t-nT) expression is by the pulse signal of pulse shaping filter shaping.

8. frequency offset estimation methods as claimed in claim 6, wherein, the complex signal S that in postponing step, has postponed the scheduled time ₂(t) obtain by following formula:

$S_2\left(t\right)=B(t-D)e^{j(2{\mathrm{\Δ\ω}}_{C}t-2{\mathrm{\Δ\ω}}_{C}D)}$

9. frequency offset estimation methods as claimed in claim 6 wherein, produces the conjugate complex signal S that produces in the step in secondary signal ₂ ^*(t) obtain by following formula:

${S_2}^{*}\left(t\right)=B(t-D)e^{-j(2{\mathrm{\Δ\ω}}_{C}t-2{\mathrm{\Δ\ω}}_{C}D)}$

10. frequency offset estimation methods as claimed in claim 6 wherein, makes it become the signal S of constant according to the phase component of complex signal ₃(t) obtain by following formula, and detect step from phase component 2 Δ ω _CDetect Δ ω among the D _C:

$S_3\left(t\right)=S_1\left(t\right)\·{S_2}^{*}\left(t\right)=B^2\left(t\right)\·e^{j2{\mathrm{\Δ\ω}}_{C}D}$

Images