Patent Publication Number: US-2015063495-A1

Title: Soft detection of m-ary dpsk signals

Description:
FIELD 
     This disclosure relates to soft detection of M-ary DPSK signals. 
     BACKGROUND 
     In wireless and wireline communications, information transmission is often accomplished by sending from the transmitter to the receiver a high-frequency carrier signal that is modulated by the message bits and transmitted over, for example, radio frequency channels, copper wires, optical fibers, cables, or any other appropriate media. Various modulation schemes may be used for signal transmission. Example modulation schemes include amplitude-shift keying (ASK), phase-shift keying (PSK), frequency-shift keying (FSK), quadrature amplitude modulation (QAM), and variations of these or other modulation schemes. 
     Additionally, differential phase-shift keying (DPSK) modulation may not require the use of pilot signals in the data stream and thus saves overhead. DPSK modulation has been employed in a wide range of scenarios, including Bluetooth and IEEE communications standards. However, conventional DPSK methods may have various drawbacks, including limited bit-error-rate performance, limited data throughput, and limited battery life. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is an example mobile communication system. 
         FIG. 2  is a schematic illustrating an example network node. 
         FIG. 3  is a schematic illustrating an example user-equipment device. 
         FIGS. 4A and 4B  are example transmitter and receiver, respectively, for wireless communications using M-ary differential PSK (DPSK) modulation and channel coding. 
         FIG. 5  is a module of generation of M-ary DPSK signal. 
         FIG. 6  is a schematic illustrating an example of hard detection of M-ary DPSK signal. 
         FIG. 7  is a schematic illustrating an example baseband model of a communication system using DPSK modulation with low-density parity check (LDPC) coding. 
         FIGS. 8-11  are schematics illustrating example receiver structures for soft detection of M-ary DPSK signals. 
         FIG. 12  is a plot illustrating example bit error rate (BER) performances of Binary-DPSK with (672,336)-LDPC under an additive white Gaussian noise (AWGN) Channel. 
         FIG. 13  is a plot illustrating example BER performances of Quaternary-DPSK and 8-DPSK with (672,336)-LDPC under AWGN Channel. 
         FIG. 14  is a flowchart illustrating an example process for soft-detection of M-ary DPSK signals. 
     
    
    
     DETAILED DESCRIPTION 
     The present embodiments include a method and system that estimate the unknown phase offset embedded in a received signal. The obtained estimate may then be either used to remove the phase offset or exploited in calculation of the log-likelihood ratio (LLR), the soft value for each bit, based on a newly derived unified expression. The new solution may be implemented in software, hardware, or embedded in a chipset. The present embodiments may improve BER performance, increase data throughput, and increase battery life as compared to conventional techniques. 
     In one aspect, the phase offset (PO) of a current signal, or portion thereof, may be estimated with respect to the previous signal, or portion thereof, received by the receiver. Then that estimated phase offset may be used in a log-likelihood ratio (LLR) calculation. As an example, an estimate of the unknown phase offset may be derived from an approximate maximum-likelihood estimation (MLE). N received symbols may be used in the MLE, which assumes that the phase offset is invariant during the period of the N symbols. Then, the LLR may be calculated using the estimated phase offset. 
     More generally, the present disclosure is directed to soft detection of DPSK signals. In many applications of wireless communications, transmitted signals propagate through fading channels and experience random variations on both signal magnitude and signal phase, which may make recovering the information carried by the signals difficult. Phase-shift keying (PSK) modulation scheme, a constant envelop modulation technique, may be robust to random variations in the magnitude, since none of the information is carried by the signal magnitude. However, PSK may be sensitive to the random phase offset (PO), since all the information is carried by absolute phase of the signal. To recover the information in a PSK receiver, the receiver may need a phase reference (and/or phase synchronization) and may use extra structural and computational complexity. In some instances, even in a static channel, acquiring a strict phase reference without phase ambiguity in a receiver may require additional training symbols to be transmitted. 
     In some instances, a PSK signal may be vulnerable to imperfect frequency synchronization between the transmitter and the receiver. For example, even a small residual frequency offset could lead to accumulated phase changes (called phase drifting) along the symbols in a packet and cause phase distortion of the received signal. In turn, the symbol recovery may be impaired and system performance may degrade. To compensate the phase drifting, training symbols are often (for example, periodically) inserted among the transmitted data symbols at the transmitter and the receiver may execute phase tracking algorithms. This approach may reduce efficiency and increase complexity. 
     To alleviate use of strict phase synchronization, differential PSK (DPSK) modulation may be used. In DPSK, the information is not carried by the absolute phase, but by the relative phase, i.e., the phase difference between a currently received symbol and a previously received symbol. Thus, in DPSK, the receiver may use the phase of the previously received signal as the reference and does not need to know the signal&#39;s absolute phase, as long as the phase offset has no significant variation during two contiguous symbol intervals. This approach may simplify the receiver complexity and make DPSK robust to the unknown random and/or slowly varying phase offset. 
     In general, DPSK may be of M-ary DPSK with M≧2, where M=2 m , m being a positive integer, with each symbol carrying m bits of information. When m=1, the M-ary DPSK becomes binary DPSK (BDPSK). DPSK may be employed in various wireless and wireline communications and networks that include, for example, Bluetooth communications, wireless local area network (WLAN) communications, millimeter wave (mmWave) communications, optical fiber communications, and others. As one example, version 2.0+ Bluetooth Enhanced Data Rate (EDR) standard and version 3.0 EDR use π/4-QDPSK and octal DPSK (M=8). By adopting these DPSK schemes, the peak data rate of Bluetooth may increase to 2 Mb/sec. and 3 Mb/sec. from 1 Mb/sec. of its previous version, respectively. As another example, IEEE 802.11ad standard for Gb/sec. short-range data communications in the unlicensed 60 GHz RF band uses the binary DPSK for the control channel. DPSK may be applied to other types of communications and be adopted in other protocol or standards. 
     In modern communications systems, modulation schemes are typically used together with error-correcting channel coding techniques to enhance the reliability of signal transmission. Various channel codes may be used, for example, convolutional code, turbo code or low-density parity check (LDPC) code. In general, the demodulator in the receiver may perform hard detection (HD) or soft detection (SD) on the modulating bits, depending on the used channel coding technique and design preferences. For instance, HD may be performed in applications where none or simple channel coding is applied, or the receiver simplicity takes precedence over achieving the best system performance. On the other hand, when the advanced iterative coding techniques such as turbo code or LDPC code are employed, the receiver may perform SD (e.g., exclusively) to supply a soft detection output (e.g., log-likelihood ratio (LLR)) for every bit to the channel decoder. 
     As mentioned above, random phase offset may be embedded in the received signal. The phase offset is generally unknown upon the receipt of the signal. Since DPSK signal does not rely upon the absolute phase of the received signal and may be robust to the phase offset, existing DPSK receivers do not estimate, identify, or otherwise exploit the phase offset of the received signal. The phase offset remains unknown during symbol detection and channel decoding at the receiver. In some implementations of the present disclosure, the phase offset embedded in the received DPSK signals may be used when calculating SD outputs (e.g., LLR) in M-ary DPSK receivers. For example, the phase offset may be estimated and the estimated phase offset may be used in LLR calculation. 
     Using the phase offset in soft detection of DPSK signals as described in this disclosure may provide one or more advantages. For example, decoding errors and the number of retransmissions may be reduced, and better error performance may be achieved. Furthermore, less transmission power may be used or the system throughput may increase (e.g., by increasing the order of modulation), without compromising the error performance. Other advantages and benefits may also be achieved. 
     I. Exemplary Communication Systems 
       FIG. 1  is an example mobile communication system  100 . In the illustrated implementations, the mobile communication system  100  includes a transmitter  102  communicably coupled to a receiver  104  through a wireless network  106 . In some implementations, the transmitter  102  may be allocated a radio resource in the wireless network  106  and transmit a radio frequency (RF) signal to the receiver  104  using the allocated resource (e.g., time, frequency bandwidth, etc.). The transmitted signal may include one or more of data packets, control signals, or other information. A data packet may be one or more transmissions. The transmitted signal may be encoded using a channel coding (e.g., turbo code, LDPC code) and modulated using a modulation scheme (e.g., PSK, DPSK, ASK, FSK, QAM). The transmitter  102  may then transmit the modulated signal via one or more antennas to the air. The receiver  102  may receive the RF signal from the air, demodulate and decode the received signal according to the modulation and channel coding schemes used at the transmitted  102  to recover the transmitted information. The transmitter  102  may communicate to the receiver  104  using any appropriate protocol stack at any appropriate time, for example, sequentially, periodically, simultaneously, or in another manner. 
     The transmitter  102  may be any electronic device configured to wirelessly transmit, for example, within the mobile communication system  100 . The transmitter  102  may transmit voice data, video data, user data, application data, multimedia data, text, web content, or any other content. In some instances, the data may be properly processed according to a modulation and a channel coding scheme before transmission. As a specific example, the transmitter may apply LDPC coding and the encoded data stream is used to generate M-ary DPSK signal. The M-ary DPSK signal may then be up-converted to a carrier frequency to be transmitted as an RF signal over the air. In some implementations, the transmitter  102  may be allocated a radio resource from the receiver  104  or the wireless network  106 . For example, the transmitter  102  may receive a broadcast of radio-resource assignments or availability of radio resources including the radio resource and associated selection criteria for transmitter  102 . In some implementations, the assignments may be transmitted using dedicated signals (e.g., control signals). Regardless, the assignments or availability of radio resources may include or otherwise identify at least one of shared radio resources, associated selection criteria, preambles, or locations of preambles within a payload. 
     The receiver  104  may be any electronic device configured to wirelessly receive, for example, within the mobile communication system  100 . For instance, the receiver  104  may receive an RF signal from one or more antennas and down-convert the received RF signal to a baseband signal for signal processing. With the baseband signal, the receiver  104  may perform demodulation and decoding to recover the transmitted information. In some instances, the receiver  104  may perform soft detection to calculate a soft detection output (e.g., log-likelihood ratio (LLR)) and use the soft detection output as the input of channel decoding to recover information transmitted from the transmitter  102 . The receiver  104  may receive voice data, video data, user data, application data, multimedia data, text, web content, or any other content. In some implementations, the receiver  104  is configured to receive, from the transmitter  102 , user data with varying transmission delays transmitted over the radio resource with varying resource identities. The receiver  104  may receive multiple transmissions in a shared resource and separate the multiple transmissions using, for example, at least one of multi-user detection (MUD), successive interference cancellation (SIC), based on maximum likelihood detection (MLD) criteria or in general any other optimization criteria. 
     In regard to a general description of the system  100 , the transmitter  102  or the receiver  104  may be user equipment (UE), a network node, or any other device in the mobile communication system  100 . For user equipment, the transmitter  102  or the receiver  104  may be referred to as mobile electronic device, user device, mobile station, subscriber station, portable electronic device, mobile communications device, wireless modem, or wireless terminal. The term “UE” may also refer to any hardware or software component that may terminate a communication session for a user. In addition, the terms “user equipment,” “UE,” “user equipment device,” “user agent,” “UA,” “user device,” and “mobile device” may be used synonymously herein. 
     Examples of user equipment may include a cellular phone, personal data assistant (PDA), smart phone, laptop, tablet personal computer (PC), pager, portable computer, portable gaming device, wearable electronic device, or other mobile communications device having components for communicating voice or data via a mobile communication network. Other examples include, but are not limited to, a television, a remote controller, a set-top box, a computer monitor, a computer (including a tablet, a desktop computer, a handheld or laptop computer, a netbook computer), a microwave, a refrigerator, a stereo system, a cassette player or recorder, a DVD player or recorder, a CD player or recorder, a VCR, an MP3 player, a radio, a camcorder, a camera, a digital camera, a portable memory chip, a washer, a dryer, a washer/dryer, a copier, a facsimile machine, a seamier, a multi-functional peripheral device, a wristwatch, a clock, and a game device, etc. The transmitter  102  or the receiver  104  may include a device and a removable memory module, such as a Universal Integrated Circuit Card (UICC) that includes a Subscriber Identity Module (SIM) application, a Universal Subscriber Identity Module (USIM) application, or a Removable User Identity Module (R-UIM) application. Alternatively, the transmitter  102  or the receiver  104  may include the device without such a module. 
     For the network node, the transmitter  102  or the receiver  104  may be referred to a base station, an access node, an access device, a relay node, a Universal Terrestrial Radio Access Network (UTRAN) node B, an eNB of an evolved Universal Terrestrial Radio Access Network (E-UTRAN), a network element, or a network component. In some implementations, in addition to wireless communications, the transmitter  102  or the receiver  104  are capable of wireline communications. For example, the transmitter  102  or the receiver  104  may be connected to a core network via a backhaul link, including an optical fiber or cable. In some instances, the transmitter  102  or the receiver  104  may be connected with each other via wirelines including copper wires, optical fiber, cables. The transmitter  102  or the receiver  104  may communicate with each other in an “Ad Hoc” mode. 
     The wireless network  106  may communicate based on orthogonal frequency division multiplexing (OFDM), orthogonal frequency division multiple access (OFDMA), space-division multiplexing (SDM), frequency-division multiplexing (FDM), time-division multiplexing (TDM), code division multiplexing (CDM), or others. The wireless network  106  may transmit information using MAC and PHY layers. Communications within the wireless network  106  may be transmitted in accordance with Long Term Evolution (LTE), Global System for Mobile Communication (GSM) protocols, Code Division Multiple Access (CDMA) protocols, Universal Mobile Telecommunications System (UMTS), Unlicensed Mobile Access (UMA), direct device-to-device (DD2D) protocols, or others. 
       FIG. 2  is a schematic illustrating an example network node  200 . As mentioned with regard to  FIG. 1 , the network node  200  may be an example of the transmitter  102  or the receiver  104 . The network node  200  may transmit and receive signals that use various modulation and channel coding schemes. As an example, the network node  200  may transmit and receive an M-ary DPSK signal with LDPC or turbo coding. The example network node  200  includes a processing module  202 , a wired communication subsystem  204 , and a wireless communication subsystem  206 . The processing module  202  may include one or more processing components (alternatively referred to as “processors” or “central processing units” (CPUs)) operable to execute instructions associated with one or more of the processes, steps, or actions described above in connection with one or more of the implementations disclosed herein (for example, the operations described with respect to  FIGS. 4-14 ). The processing module  202  may also include other auxiliary components, such as random access memory (RAM), read only memory (ROM), or secondary storage (for example, a hard disk drive or flash memory). The processing module  202  may execute certain instructions and commands to provide wireless or wired communication, using the wired communication subsystem  204  or a wireless communication subsystem  206 . A skilled artisan will readily appreciate that various other components may also be included in the example network node  200 . 
       FIG. 3  is a schematic illustrating an example UE  300 . As mentioned with regard to  FIG. 1 , the UE  300  may be an example of the transmitter  102  or the receiver  104 , The UE  300  may transmit and receive signals that use various modulation and channel coding schemes. As an example, the UE  300  may transmit and receive an M-ary DPSK signal with LDPC or turbo coding. The example UE  300  includes a processing unit  302 , a computer readable storage medium  304  (for example, ROM or flash memory), a wireless communication subsystem  306 , an interface  308 , and an I/O interface  310 . Similar to the processing module  202  of  FIG. 2 , the processing unit  302  may include one or more processing components (alternatively referred to as “processors” or “central processing units” (CPUs)) configured to execute instructions related to one or more of the processes, steps, or actions described above in connection with one or more of the implementations disclosed herein (for example, operations described with respect to  FIGS. 4-14 ). The wireless communication subsystem  306  may be configured to provide wireless communications for data information or control information provided by the processing unit  302 . The wireless communication subsystem  306  may include, for example, one or more antennas, a receiver, a transmitter, a local oscillator, a mixer, and a digital signal processing (DSP) unit. In some embodiments, the wireless communication subsystem  306  may support advanced multi-user detection (MUD) receivers and multiple input multiple output (MIMO) transmissions. 
     The interface  308  may include, for example, one or more of a screen or touch screen (for example, a liquid crystal display (LCD), a light emitting display (LED), an organic light emitting display (OLED), a microelectromechanical system (MEMS) display), a keyboard or keypad, a trackball, a speaker, and a microphone. The I/O interface  310  may include, for example, a universal serial bus (USB) interface. A skilled artisan will readily appreciate that various other components may also be included in the example UE device  300 . 
     II. Exemplary Transmitter and Receiver 
       FIG. 4A  is an example transmitter  410  for wireless communications using M-ary differential PSK (M-DPSK) modulation and channel coding. The example transmitter  410  may be, for example, the transmitter  102  in  FIG. 1 . In some instances, the example transmitter  410  may be included in the example network node  200  in  FIG. 2 , the example UE  300  in  FIG. 3 , or any other appropriate device. As illustrated, the example transmitter  410  includes a baseband processing subsystem  430 , a radio frequency (RF) circuit subsystem  435 , and one or more antennas  417 . The baseband processing subsystem  430  may include any appropriate software, hardware, firmware, or a combination thereof for baseband signal processing including, for example, modulation, channel encoding, or others. As illustrated, the baseband processing subsystem  430  includes a channel encoder  411 , a differential encoder  412 , an M-ary PSK modulator  413 , and a digital-to-analog (D/A) converter  414 . The baseband processing subsystem  430  may include additional or different components without departing from the scope of this disclosure. The RF circuit subsystem  435  may include any appropriate software, hardware, firmware, or a combination configured for RF processing including, for example, up-converting baseband signal to radio frequency, power amplification, or others. As illustrated, the RF circuit subsystem  435  includes an up-converter  415 , and a power amplifier  416 . The RF circuit subsystem  435  may include additional or different components without departing from the scope of this disclosure. 
     In some implementations, information bits may be input into the channel encoder  411  for channel coding. The channel encoder  411  may be configured to execute, for example, channel coding algorithms of convolutional code, turbo code, LDPC code, or any other appropriate channel code. In some instances, the channel encoder  411  may be referred to as an error-correcting encoder. After the channel encoding, the encoded bits are passed to the differential encoder  412  to perform differential coding. The differentially encoded bit stream is then applied to the M-ary PSK modulator  413  for phase-shift keying (PSK) modulation. The combination of the differential encoder  412  and the Mary PSK modulator  413  acts as an example of an M-ary DPSK modulator. The M-ary DPSK modulated symbols may be converted to analog signals by the digital-to-analog (D/A) converter  414 , resulting in an analog baseband signal. The analog baseband signal may then be, via up-converter  415 , tuned from baseband to radio frequency (RF) band (for example, a carrier frequency band assigned to the transmitter  410  for transmission). The RF signal may be amplified by the amplifier  416  and transmitted by one or more antennas  417  to the air. The RF signal may experience path loss, amplitude and phase variations, distortions, or a combination thereof during propagation over the RF channel and eventually arrive at a receiver. 
       FIG. 4B  is an example receiver  420  for wireless communications using M-ary differential PSK modulation and channel coding. The example transmitter  420  may be, for example, the receiver  104  in  FIG. 1 . In some instances, the example receiver  420  may be included in the example network node  200  in  FIG. 2 , the example UE  300  in  FIG. 3 , or any other appropriate device. As illustrated, the example receiver  420  includes a RF circuit subsystem  440 , a baseband processing subsystem  445 , and one or more antennas  421 . The RF circuit subsystem  440  may include any appropriate software, hardware, firmware, or a combination configured for RF processing including, for example, down-converting RF signal to baseband, power amplification, or other operations. For instance, the illustrated RF circuit subsystem  440  may include one or more antennas  421 , a band-pass filter (BPF)  422 , an automatic gain controller (AGC)  423 , and a down-converter  424 . The RF circuit subsystem  440  may include additional or different components without departing from the scope of this disclosure. The baseband processing subsystem  445  may include any appropriate software, hardware, firmware, or a combination of these and other elements and be configured for baseband processing (e.g., demodulation, channel decoding, signal detection, etc.). For instance, the illustrated baseband processing subsystem  445  includes an analog-to-digital (A/D) converter  425 , a log-likelihood ratio calculator  426 , and a channel decoder  427 . The baseband processing subsystem  445  may include additional or different components without departing from the scope of this disclosure. 
     In some implementations, the one or more antennas  421  receive RF signals from the air. The received RF signal may pass through the BPF  422  and the AGC  423  to filter out the out-of-hand noise and interference and to amplify the signal to a specified magnitude level, respectively. The down-convertor  424  down converts the processed RF signal to the baseband. The A/D converter  425  converts the received baseband signal to a digital signal, and the digital signal may then be demodulated and decoded. For example, the received digital signal may be passed to the LLR calculator  426  for DPSK demodulation with soft detection. The output of the LLR calculator  426  may be passed to the channel decoder  427  to recover the transmitted information bits. The channel decoder  427  may perform a channel decoding algorithm matched to the channel code (e.g., convolutional code, turbo code, LDPC code) used on the transmitter side. 
     III. Exemplary M-DPSK Techniques 
       FIG. 5  is a module  500  for generating M-ary DPSK (M-DPSK) signals. The module  500 , as illustrated, includes a serial-to-parallel (SIP) converter  510 , a differential encoder  520 , a delay  530 , and an M-PSK modulator  540 . In some implementations, the S/P converter  510  receives a sequence of channel-coded bits and divides the sequence into multiple groups, where each group may include m=log 2 M channel-coded bits. Without loss of generality, in this disclosure, each bit may take a value of 1 or −1 with an equal probability. In cases where each bit takes a value of 1 or 0, or any other binary values, they may always be simply converted to 1 or −1. The baseband M-DPSK signal for the k-th symbol may be expressed as 
         x   k =√{square root over (P s )}e jφk , for  k= 0,1,2, . . .   (1)
 
     where j=√{square root over (−1)}, P s  is the transmitted signal power and φ k  is the phase of the k-th symbol. While in PSK, the information of the bits is carried over directly by φ k ; in DPSK it is carried over by the difference between φ k  and φ k-1 , i.e., by δ k =φ k −φ k-1 . Thus, 
       φ k =φ k-1 +δ k , for  k= 1,2, . . .   (2)
 
     where φ 0  may be predetermined. For example, φ 0  may be equal to zero for the binary case and zero, π/M, or another value for M&gt;2. In Equation (2), δ k  may take a value from an M-ary set  , for instance, 
       δ k ε ≡{2 lπ/M:l= 0,1 , . . . ,M− 1}.  (3)
 
     δ k  may represent the k-th group of m channel-coded bits, b k,i ξ{−1,1}, i=1, 2, . . . , m, according to a predetermined mapping rule. Tables 1-3 show examples of the mapping rules for M=2, 4 and 8 respectively. From Equations (1) and (2), the k-th symbol may be expressed as 
         x   k   =x   k-1   e   jδ     k   , for  k= 1,2, . . .   (4)
 
     Equation (4) indicates that the symbol transmitted in the k-th interval is determined by the output of the differential encoder  520  based on both the k-th group of the channel-coded bits (output form the S/P converter  510 ) and the previous symbol x k-1  (output from the delay  530 ), as illustrated by  FIG. 5 . It may be seen that in Equation (1), the phase of each symbol, φ k , belongs to an M-ary set  , for example, 
       φ k ε ≡{2 lπ/M+φ   0   :l= 0,1 , . . . ,M− 1}.  (5)
 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Example Mapping Rule for M = 2 
               
            
           
           
               
               
               
            
               
                 b k   
                 δ k   
                 n 
               
               
                   
               
            
           
           
               
               
               
            
               
                 1 
                 0 
                 0 
               
               
                 −1 
                 π 
                 1 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Example Mapping Rule for M = 4 
               
            
           
           
               
               
               
               
               
            
               
                   
                 b k, 2   
                 b k, 1   
                 δ k   
                 n 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 1 
                 1 
                 0 
                 0 
               
               
                   
                 1 
                 −1 
                  π/2 
                 1 
               
               
                   
                 −1 
                 −1 
                 π 
                 2 
               
               
                   
                 −1 
                 1 
                 3π/2 
                 3 (or −1) 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Example Mapping Rule for M = 8 
               
            
           
           
               
               
               
               
               
            
               
                 b k, 3   
                 b k, 2   
                 b k, 1   
                 δ k   
                 n 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 1 
                 1 
                 1 
                 0 
                 0 
               
               
                 −1 
                 1 
                 1 
                  π/4 
                 1 
               
               
                 −1 
                 1 
                 −1 
                  π/2 
                 2 
               
               
                 −1 
                 −1 
                 −1 
                 3π/4 
                 3 
               
               
                 −1 
                 −1 
                 1 
                 π 
                 4 
               
               
                 1 
                 −1 
                 1 
                 5π/4 
                 5 (or −3) 
               
               
                 1 
                 −1 
                 −1 
                 3π/2 
                 6 (or −2) 
               
               
                 1 
                 1 
                 −1 
                 7π/4 
                 7 (or −1) 
               
               
                   
               
            
           
         
       
     
     In Tables 1-3, the minimum absolute value (MAV) of δ k , denoted by φ≡min |δ k |, is zero. The MAV of δ k  may be another non-zero value. For example, the quatemary DPSK (QDPSK) with φ=π/4 may be referred to as π/4-QDPSK. Table 4 lists an example mapping rule for π/4-QDPSK. Equivalently, π/4-QDPSK may be viewed as a regular QDPSK (with φ=0) plus a continuing phase rotation of π/4 from each symbol to the next symbol. 
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 Example Mapping Rule for π/4-QDPSK 
               
            
           
           
               
               
               
               
               
            
               
                   
                 b k, 2   
                 b k, 1   
                 δ k   
                 n 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 1 
                 1 
                  π/4 
                 0 
               
               
                   
                 1 
                 −1 
                 3π/4 
                 1 
               
               
                   
                 −1 
                 −1 
                 5π/4 
                 2 
               
               
                   
                 −1 
                 1 
                 7π/4 
                 3 (or −1) 
               
               
                   
                   
               
            
           
         
       
     
     After the modulated QDPSK signal propagates through an additive white Gaussian noise (AWGN) channel or a flat fading channel, the received k-th symbol at a receiver may be expressed as 
       {hacek over ( r )} k   =αe   jθ   x   k +{hacek over ( n )} k =α√{square root over ( P   s )}e j(φ     k     +θ)   +{hacek over (n)}   k ,  (6)
 
     where α&gt;0 stands for the magnitude propagation coefficient and θ represents an unknown phase offset embedded in the received signal. For the AWGN channel, both α and θ may be constant. For a block fading channel (also called quasi-static fading channel), both α and θ are random variables (RV&#39;s) varying independently from one block of N symbols to another block, but may be assumed to be invariant in each block of N symbols. In Equation (6), {hacek over (n)} k  represents the noise. In general, {hacek over (n)} k  may be a complex AWGN noise, of which the real and imaginary components are independent zero-mean Gaussian random variables, each with a variance of {hacek over (σ)} 2 . {hacek over (n)} k  is assumed to be independent of {hacek over (n)} k , for k≠k′. In some implementations, the received signal is normalized. The received signal after normalization may be expressed as 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       k 
                     
                     = 
                     
                       
                         
                           
                             r 
                             ⋓ 
                           
                           k 
                         
                         
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                               P 
                               s 
                             
                           
                         
                       
                       = 
                       
                         
                            
                           
                             j 
                              
                             
                               ( 
                               
                                 
                                   φ 
                                   k 
                                 
                                 + 
                                 θ 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           n 
                           k 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     n 
                     k 
                   
                   = 
                   
                     
                       
                         n 
                         ⋓ 
                       
                       k 
                     
                     
                       α 
                        
                       
                         
                           P 
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     is a scaled version of {hacek over (n)} k . The variance of each of the real and the imaginary components of n k  may be given by 
       σ 2 ={hacek over (σ)} 2 /α 2   P   s =1/2ρ,  (9)
 
     where ρ=α 2 P s /{hacek over (σ)} 2 =1/2σ 2  is the signal-to-noise ratio (SNR) of the received signal. 
       FIG. 6  is a schematic  600  illustrating an example of hard detection (HD) of M-ary DPSK signal. In some implementations, at the receiver, hard detection may be performed on a symbol-by-symbol basis to recover the channel-coded bits. As illustrated in  FIG. 6 , the k-th received symbol r k  is multiplied, at  630 , with the complex conjugate (at  620 ) of the previous received signal r k-1  (output from the delay  610 ). The receiver may compute the angle 
       {tilde over (δ)} k =∠( r   k   r   k-1 *),  (10)
 
     where * represents the complex conjugate, and ∠(.) represents the angle of the argument. The angle {tilde over (δ)} k  represents the phase difference between two consecutively received symbols. Based on {tilde over (δ)} k , a hard decision may be made at  640  such that, for example, 
       {circumflex over (δ)} k =arg min δε   |{tilde over (δ)} k −δ|,  (11)
 
     where δ belongs to the set   of δ k  as listed in the same mapping table used by the transmitter (e.g., Tables 1-4). The information bits are recovered at  650  by de-mapping {circumflex over (δ)} k  to the output bits {circumflex over (b)} k,i , i=1, 2, . . . , m. For the binary case, the HD process may be simplified to 
     
       
         
           
             
               
                 
                   
                     
                       b 
                       ^ 
                     
                     k 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             1 
                           
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 
                                   z 
                                   k 
                                 
                               
                               ≥ 
                               0 
                             
                           
                         
                         
                           
                             
                               - 
                               1 
                             
                           
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 
                                   z 
                                   k 
                                 
                               
                               &lt; 
                               0 
                             
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       where 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                     k 
                   
                   = 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       { 
                       
                         
                           r 
                           k 
                         
                          
                         
                           r 
                           
                             k 
                             - 
                             1 
                           
                           * 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     with R eal {.} standing for the real component of the argument. In other words, if the conjugate product of two consecutively received symbols has a non-negative real component, it is determined that a bit  1  is transmitted; otherwise, a bit −1 is transmitted. 
       FIG. 7  is a schematic  700  illustrating an example baseband model of a communication system using DPSK modulation with low-density parity check (LDPC) coding. In modern communications systems, modulation techniques are usually used in combination with error-correcting channel coding techniques, such as those based on the turbo code or the low-density parity check (LDPC) code. For instance, in the IEEE 802.1 lad standard, binary DPSK is adopted for the control channel accompanied by a half-rate (672, 336)-LDPC. In such applications, a soft-detection (SD) technique is used exclusively. For example, the DPSK demodulator needs to supply the channel decoder with a soft-detection output for every bit. The SD output may include, for example, likelihood ratio (LR), log-likelihood ratio (LLR), or any other appropriate metric. As illustrated in  FIG. 7 , the example baseband model includes an LDPC encoder  710 , a differential PSK modulator  720 , a channel  730 , an LLR calculator  740 , and an LDPC decoder  750 . As an example process, on the transmitter side, information bits {a k } may first be channel coded, for example, by the LDPC encoder  710  into coded bits {b k }. The coded bits {b k } may then be converted into modulated symbols {x k } (e.g., DPSK modulated symbols) via the DPSK modulator  720 . The modulated symbols {x k } may go through a channel  730  and arrive at the receiver. On the receiver side, received symbols {r k } may be input into, for example, an LLR calculator  740  for soft detection. Based on the calculated soft metrics (e.g., LLR), the LDPC decoder  750  may determine and produce output bits {{circumflex over (b)} k }. 
     A. Binary Case 
     In some implementations, an example method to calculate the LLR for Binary DPSK (BDPSK) may be based on z k  of Equation (13). Following the maximum a posteriori (MAP) principle, the LLR of the k-th bit b k  may be calculated as 
     
       
         
           
             
               
                 
                   
                     
                       LLR 
                       k 
                     
                     = 
                     
                       
                         ln 
                          
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     z 
                                     k 
                                   
                                   | 
                                   
                                     b 
                                     k 
                                   
                                 
                                 = 
                                 1 
                               
                               ) 
                             
                           
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   
                                     z 
                                     k 
                                   
                                   | 
                                   
                                     b 
                                     k 
                                   
                                 
                                 = 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         ln 
                          
                         
                           
                             
                               P 
                               r 
                             
                              
                             
                               ( 
                               
                                 
                                   b 
                                   k 
                                 
                                 = 
                                 1 
                               
                               ) 
                             
                           
                           
                             
                               P 
                               r 
                             
                              
                             
                               ( 
                               
                                 
                                   b 
                                   k 
                                 
                                 = 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     where p(z k |b k =1) is the probability density function (pdf) of z k  under the condition of b k =1, and P r (b k =1) is the probability for b k =1. In the case that b k  takes 1 or −1 with an equal probability, as assumed in analyses below, the second term on the right-hand side (RHS) of (14) becomes zero and is dropped off. In other cases, when b k  takes 1 or −1 with an unequal probability, the second term on the RHS of (14) is non-zero and should be included. In (14), the expression for the accurate pdf of z k  is quite complicated, and an accurate expression is rarely used for LLR calculation. In reality, z k  may be approximated by a Gaussian random variable, for example, when the SNR is sufficiently high. Substituting the pdf expression of the approximate Gaussian distribution for z k  into Equation (14) may yield 
         LLR   k ≈2 ρz   k ,  (15)
 
     where ρ is the signal-to-noise ratio. In some instances, the above obtained LLR based on Gaussian approximation (GA) of z k  may not be optimal, in a sense that the resultant bit error (BER) is not the minimum. 
     In some other implementations, LLR, may be calculated based on the joint pdf of r k  and r k-1 , rather than on the pdf of z k . For example, the LLR of the k-th bit b k  may be calculated as 
     
       
         
           
             
               
                 
                   
                     LLR 
                     k 
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 r 
                                 k 
                               
                               , 
                               
                                 
                                   
                                     r 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                   | 
                                   
                                     b 
                                     k 
                                   
                                 
                                 = 
                                 1 
                               
                             
                             ) 
                           
                         
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 r 
                                 k 
                               
                               , 
                               
                                 
                                   
                                     r 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                   | 
                                   
                                     b 
                                     k 
                                   
                                 
                                 = 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Alternatively or differently, two other expressions for LLR calculation may be used, for example, 
     
       
         
           
             
               
                 
                   
                     
                       LLR 
                       k 
                     
                     = 
                     
                       ln 
                        
                       
                         
                           
                             l 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 ρ 
                               
                               | 
                               
                                 
                                   r 
                                   k 
                                 
                                 + 
                                 
                                   r 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               | 
                             
                             ) 
                           
                         
                         
                           
                             l 
                             0 
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 ρ 
                               
                               | 
                               
                                 
                                   r 
                                   k 
                                 
                                 - 
                                 
                                   r 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               | 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       LLR 
                       k 
                     
                     = 
                     
                       ln 
                        
                       
                         
                           cosh 
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 ρ 
                               
                               | 
                               
                                 
                                   r 
                                   k 
                                 
                                 + 
                                 
                                   r 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               | 
                             
                             ) 
                           
                         
                         
                           cosh 
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 ρ 
                               
                               | 
                               
                                 
                                   r 
                                   k 
                                 
                                 - 
                                 
                                   r 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               | 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     where cos h(x)≡(e x +e −x )/2 and I 0 (.) is the zero-order modified Bessel function of the first kind. In some instances, soft detection based on Equations (15), (17) and (18) may not differ much in their error rate performances. In some implementations, iterative approaches for LLR calculation may be used and such iterative approaches may involve extra complexity and time delay. 
     B. M-ary Case 
     For the more general cases of M-ary DPSK with M&gt;2, the calculation of LLR may be more complicated. As an example, Equation (14) may be applied to the M-ary cases with z k  replaced by y k , where 
         y   k   =r   k   r   k-1 *.  (19)
 
     y k  is the conjugate product of two consecutively received symbols r k  and r k-1 . In some instances, when SNR is sufficiently high, y k  may be well approximated by a complex Gaussian RV with a mean value equal to e jδ     k   , where δ k  may have a predetermined mapping relationship (e.g., as in Tables 1-4) with the bits b k,i , i=1˜m. As a result, for the bit b k,i , an approximate LLR expression based on GA may be derived as 
         LLR   k,i ≈ρ[max nεb     k,i     =1   R   eal ( y   k   e   −j2nπ/M )−max nεb     k,i     =−1   R   eal ( y   k   e   −j2nπ/M )],  (20)
 
     where n in each term inside the square brackets may be determined by the mapping rule between δ k  and b k,i  where b k,i  equals to 1 or −1. As an instance, according to the mapping rule in Table 2, for r=1, the values of n satisfying nεb k,1 =1 are 0 and −1 (or 3) while for t=2, the values of n satisfying nεb k,2 =1 are 0 and 1. When M=2, Equation (20) reduces to Equation (15) of the binary case. 
     C. Exemplary Receiver Structures 
       FIG. 8  is a schematic illustrating an example receiver structure  800  for soft detection of DPSK signals. The example receiver structure  800  may, for example, implement the LLR calculation based on Equation (15) or (20) which may involve conjugate multiplication of two DPSK symbols. As illustrated, a received signal (e.g., a radio frequency signal) may first be down-converted (e.g., by the clown-convertor  424  in  FIG. 4B ) to a baseband signal, which may be further sampled into a digital signal. The received symbols {r k } may go through a conjugate multiplication process (e.g., via a delay  810 , a complex conjugate operator  820 , and a multiplier unit  830 ) and obtain the conjugate product y k  of two consecutively received symbols r k  and r k-1 . At  840 , based on the conjugate product y k , LLR may be calculated according to Equation (15) or (20). The channel decoder  850  may decode the information bits based on the calculated LLR values output from the LLR calculator. 
       FIG. 9  is a schematic illustrating another example receiver structure  900  for soft detection of DPSK signals. The example receiver structure  900  may, for example, implement the soft detection of DPSK signals based on Equation (17) or (18). As illustrated, two consecutively received baseband symbols r k  and r k-1  may be directly supplied into an LLR calculator  920  for LLR calculation according to Equation (17) or (18). The output LLR values are used to decode information bits by a channel decoder  930 . 
     The example methods described above with respect to Equations (15)-(18) for soft detection of DPSK signals do not exploit the phase offset (PO) embedded in the received signal. As a result, their resulting bit-error rate may be far from the minimum. In some implementations, the PO of the received DPSK signal may be identified and exploited for soft detection of DPSK signals. For example, the receiver may estimate the PO of the received signal and use the estimated PO in LLR calculation. The bit error performance may be improved by exploiting the PO in soft detection of DPSK signals. 
     IV. Exemplary Method 
     In some implementations, the PO may be estimated, for example, based on the received DPSK signal according to maximum likelihood (ML) principle. For example, denote {circumflex over (θ)} as an estimate of the unknown PO of the received signal. {circumflex over (θ)} may be derived from an approximate maximum-likelihood estimation (MLE) as, 
     
       
         
           
             
               
                 
                   
                     
                       θ 
                       ^ 
                     
                     = 
                     
                       
                         
                           1 
                           M 
                         
                          
                         atan 
                          
                         
                           
                             
                               I 
                               mag 
                             
                              
                             
                               ( 
                               
                                 
                                   Σ 
                                   
                                     k 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                  
                                 
                                   r 
                                   k 
                                   M 
                                 
                               
                               ) 
                             
                           
                           
                             
                               R 
                               eal 
                             
                              
                             
                               ( 
                               
                                 
                                   Σ 
                                   
                                     k 
                                     = 
                                     1 
                                   
                                   N 
                                 
                                  
                                 
                                   r 
                                   k 
                                   M 
                                 
                               
                               ) 
                             
                           
                         
                       
                       + 
                       
                         2 
                          
                         λπ 
                          
                         
                           / 
                         
                          
                         M 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     where a tan stands for arctangent function, I mag (.) stands for the imaginary component of the argument, and λ may be any integer. As indicated by (21), N received symbols may be used in the MLE, assuming the phase offset is invariant during the period of these N symbols. The LLR may be calculated, for example, using the following expression, 
     
       
         
           
             
               
                 
                   
                     LLR 
                     k 
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           Σ 
                           
                             
                               n 
                               ∈ 
                               
                                 b 
                                 
                                   k 
                                   , 
                                   i 
                                 
                               
                             
                             = 
                             1 
                           
                         
                          
                         
                           Σ 
                           
                             l 
                             = 
                             0 
                           
                           
                             
                               M 
                                
                               
                                 / 
                               
                                
                               2 
                             
                             - 
                             1 
                           
                         
                          
                         
                           cosh 
                            
                           
                             ( 
                             
                               2 
                                
                               ρ 
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   eal 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           r 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         + 
                                         
                                           
                                             r 
                                             k 
                                           
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 j 
                                               
                                                
                                               
                                                 
                                                   2 
                                                    
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                 
                                                 M 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         - 
                                         
                                           j 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   2 
                                                    
                                                   l 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                 
                                                 M 
                                               
                                               + 
                                               
                                                 θ 
                                                 ^ 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           Σ 
                           
                             
                               n 
                               ∈ 
                               
                                 b 
                                 
                                   k 
                                   , 
                                   i 
                                 
                               
                             
                             = 
                             
                               - 
                               1 
                             
                           
                         
                          
                         
                           Σ 
                           
                             l 
                             = 
                             0 
                           
                           
                             
                               M 
                                
                               
                                 / 
                               
                                
                               2 
                             
                             - 
                             1 
                           
                         
                          
                         
                           cosh 
                            
                           
                             ( 
                             
                               2 
                                
                               ρ 
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   eal 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           r 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         + 
                                         
                                           
                                             r 
                                             k 
                                           
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 j 
                                               
                                                
                                               
                                                 
                                                   2 
                                                    
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                 
                                                 M 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         - 
                                         
                                           j 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   2 
                                                    
                                                   l 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                 
                                                 M 
                                               
                                               + 
                                               
                                                 θ 
                                                 ^ 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     22 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       = 
                       
                         ln 
                          
                         
                           
                             
                               Σ 
                               
                                 
                                   n 
                                   ∈ 
                                   
                                     b 
                                     
                                       k 
                                       , 
                                       i 
                                     
                                   
                                 
                                 = 
                                 1 
                               
                             
                              
                             
                               Σ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 
                                   M 
                                    
                                   
                                     / 
                                   
                                    
                                   2 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               cosh 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   ρ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       R 
                                       eal 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               
                                                 k 
                                                 - 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               
                                                 
                                                   r 
                                                   ^ 
                                                 
                                                 k 
                                               
                                                
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                     j 
                                                   
                                                    
                                                   
                                                     
                                                       2 
                                                        
                                                       n 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       π 
                                                     
                                                     M 
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                            
                                           
                                             
                                               - 
                                               j 
                                             
                                              
                                             
                                               
                                                 2 
                                                  
                                                 l 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                               
                                               M 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               Σ 
                               
                                 
                                   n 
                                   ∈ 
                                   
                                     b 
                                     
                                       k 
                                       , 
                                       i 
                                     
                                   
                                 
                                 = 
                                 
                                   - 
                                   1 
                                 
                               
                             
                              
                             
                               Σ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 
                                   M 
                                    
                                   
                                     / 
                                   
                                    
                                   2 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               cosh 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   ρ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       R 
                                       eal 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 r 
                                                 ^ 
                                               
                                               
                                                 k 
                                                 - 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               
                                                 
                                                   r 
                                                   ^ 
                                                 
                                                 k 
                                               
                                                
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                     j 
                                                   
                                                    
                                                   
                                                     
                                                       2 
                                                        
                                                       n 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       π 
                                                     
                                                     M 
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                          
                                         
                                            
                                           
                                             
                                               - 
                                               j 
                                             
                                              
                                             
                                               
                                                 2 
                                                  
                                                 l 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                               
                                               M 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     , 
                     
                       
                         for 
                          
                         
                             
                         
                          
                         i 
                       
                       = 
                       1 
                     
                     , 
                     2 
                     , 
                     … 
                     , 
                     m 
                     , 
                     
                       
 
                     
                      
                     where 
                   
                 
               
               
                 
                   ( 
                   
                     22 
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       r 
                       ~ 
                     
                     k 
                   
                   ≡ 
                   
                     
                       r 
                       k 
                     
                      
                     
                       
                          
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           
                             θ 
                             ^ 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     {tilde over (r)} k  may be considered as a phase-compensated version of the received r k , with a phase de-rotation based on the estimated PO {circumflex over (θ)}. In some other implementations, the PO may be estimated or otherwise identified based on another technique. 
     Note that the MLE of θ obtained from (21) has a phase ambiguity of 2λπ/M radians, λ being any integer. However, the effect of introducing a change of 2λπ/M in {circumflex over (θ)} is equivalent to a change of the second summation sign in the numerator as well as the denominator of (22a) from Σ l=0   M/2−1  to Σ l=λ   M/2−1+λ , which does not change the final LLR value. Therefore, any phase ambiguity of {circumflex over (θ)} equal to 2λπ/M radians does not affect the calculation of LLR. 
     In some instances, when the SNR is high, (22b) may be well approximated by 
     
       
         
           
             
               
                 
                   
                     
                       LLR 
                       
                         k 
                         , 
                         i 
                       
                     
                     ≈ 
                     
                       2 
                        
                       ρ 
                        
                       
                         max 
                         
                           
                             l 
                             = 
                             
                               0 
                                
                               
                                 : 
                               
                                
                               
                                 ( 
                                 
                                   
                                     M 
                                      
                                     
                                       / 
                                     
                                      
                                     2 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                            
                           
                             
 
                           
                            
                           
                             
                               n 
                               ∈ 
                               
                                 b 
                                 
                                   k 
                                   , 
                                   i 
                                 
                               
                             
                             = 
                             1 
                           
                         
                       
                     
                   
                   | 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     M 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                               
                                 2 
                                  
                                 l 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   | 
                   
                     
                       - 
                       2 
                     
                      
                     ρ 
                      
                     
                       max 
                       
                         
                           l 
                           = 
                           
                             0 
                              
                             
                               : 
                             
                              
                             
                               ( 
                               
                                 
                                   M 
                                    
                                   
                                     / 
                                   
                                    
                                   2 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                          
                         
                           
 
                         
                          
                         
                           
                             n 
                             ∈ 
                             
                               b 
                               
                                 k 
                                 , 
                                 i 
                               
                             
                           
                           = 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                   | 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     M 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                               
                                 2 
                                  
                                 l 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   | 
                   . 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Equation (24) helps simplify the calculation by avoiding evaluation of the hyperbolic function and the logarithm function, as well as the division. 
       FIG. 10  is a schematic illustrating an example receiver structure  1000  for soft detection of DPSK signals. The example receiver structure  1000  may, for example, implement the soft detection of DPSK signals based on Equations (21)-(23). As illustrated, phase offset estimation may be performed based on the received baseband symbols {r k } at  1030 . The phase offset may be estimated, for example, according to the example MLE described with respect to Equation (21), or the phase offset may be estimated in another manner, Based on the estimated PO {circumflex over (θ)}, a phase de-rotation factor e −j{circumflex over (θ)}  may be calculated at  1040 . The phase de-rotation factor e −j{circumflex over (θ)} , together with received symbols r k  and r k-1 , may be used for LLR calculation according to Equation (22a) at  1050 . The channel decoder  1060  may take the calculated LLR and determine output bits. 
       FIG. 11  is a schematic illustrating another example receiver structure  1100  for soft detection of DPSK signals. The example receiver structure  1100  may, for example, implement the soft detection of DPSK signals based on Equations (21)-(24). Similar to the example receiver structure  1000 , a phase offset estimate {circumflex over (θ)} may be determined based on the received symbols {r k } at  1120 . Each received symbol r k  may be phase-compensated based on the phase offset estimate {circumflex over (θ)} at  1140  (e.g., by multiplying the de-rotation factor e −j{circumflex over (θ)} , or subtracting the phase of r k  by {circumflex over (θ)} radians) and obtain the phase-compensated {tilde over (r)} k  as given in Equation (23). The phase-compensated received symbols, {tilde over (r)} k  and {tilde over (r)} k-1  may be supplied into the LLR calculation at  1160 , for example, based on Equation (22b) or (24). Similarly, based on the calculated LLR, a channel decoder  1170  may determine output bits. 
     The receiver may include additional or different components or may be configured in another manner to perform LLR calculation (e.g., based on Equations (21)-(24)). Note that in  FIG. 10  and  FIG. 11 , the dashed-line box  1010  and  1110  may be used in the case that the minimum absolute value (MAV) of δ k , denoted by φ, is not zero, (e.g., for the π/4-QDPSK) while being bypassed when φ=0. Specifically, when φ≠0, the k-th received symbol may be de-rotated by kφ radians (i.e., rotated by −kφ radians). Then the symbols after de-rotation may be treated as φ=0. 
     A. Binary DPSK 
     In the case of Binary DPSK (BDPSK, M=2), when the example mapping rule of Table 1 is used, Equation (22) may be simplified to 
     
       
         
           
             
               
                 
                   
                     LLR 
                     k 
                   
                   = 
                   
                     ln 
                      
                     
                       
                         cosh 
                          
                         
                           [ 
                           
                             2 
                              
                             ρ 
                              
                             
                                 
                             
                              
                             
                               
                                 R 
                                 eal 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       ~ 
                                     
                                     k 
                                   
                                   + 
                                   
                                     
                                       r 
                                       ~ 
                                     
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       
                         cosh 
                          
                         
                           [ 
                           
                             2 
                              
                             ρ 
                              
                             
                                 
                             
                              
                             
                               
                                 R 
                                 eal 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       r 
                                       ~ 
                                     
                                     k 
                                   
                                   - 
                                   
                                     
                                       r 
                                       ~ 
                                     
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     25 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     for b k . 
     Alternatively or additionally, the high SNR approximation based on Equation (24) may be applied and the LLR in (25a) may be approximated by 
         LLR   k ≈2 ρ[|R   eal ( {tilde over (r)}   k   +{tilde over (r)}   k-1 )|−| R   eal ( {tilde over (r)}   k   −{tilde over (r)}   k-1 )|].  (25b)
 
     In this case, the LLR for each bit depends on the difference between the absolute value of the real part of the sum of two consecutively received symbols with phase compensation, and the absolute value of the real part of the difference of two consecutively received symbols with phase compensation. 
     B. Quaternary DPSK 
     In the case of Quaternary DPSK (QDPSK, M=4), when the example mapping rule of Table 2 is used, Equations (22a) and (22b) (collectively, Equation (22)) may be simplified as 
     
       
         
           
             
               
                 
                   
                     
                       LLR 
                       
                         k 
                         , 
                         2 
                       
                     
                     = 
                     
                       ln 
                        
                       
                         
                           
                             Σ 
                             
                               n 
                               = 
                               0 
                             
                             1 
                           
                            
                           
                             Σ 
                             
                               l 
                               = 
                               0 
                             
                             1 
                           
                            
                           
                             cosh 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 ρ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                     eal 
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               r 
                                               ~ 
                                             
                                             
                                               k 
                                               - 
                                               1 
                                             
                                           
                                           + 
                                           
                                             
                                               
                                                 r 
                                                 ~ 
                                               
                                               k 
                                             
                                              
                                             
                                                
                                               
                                                 
                                                   - 
                                                   j 
                                                 
                                                  
                                                 
                                                   
                                                     n 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     π 
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                          
                                         
                                           
                                             - 
                                             j 
                                           
                                            
                                           
                                             
                                               l 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             Σ 
                             
                               n 
                               = 
                               2 
                             
                             3 
                           
                            
                           
                             Σ 
                             
                               l 
                               = 
                               0 
                             
                             1 
                           
                            
                           
                             cosh 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 ρ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                     eal 
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               r 
                                               ~ 
                                             
                                             
                                               k 
                                               - 
                                               1 
                                             
                                           
                                           + 
                                           
                                             
                                               
                                                 r 
                                                 ~ 
                                               
                                               k 
                                             
                                              
                                             
                                                
                                               
                                                 
                                                   - 
                                                   j 
                                                 
                                                  
                                                 
                                                   
                                                     n 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     π 
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                          
                                         
                                           
                                             - 
                                             j 
                                           
                                            
                                           
                                             
                                               l 
                                                
                                               
                                                   
                                               
                                                
                                               π 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   
                     26 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     LLR 
                     
                       k 
                       , 
                       1 
                     
                   
                   = 
                   
                     ln 
                      
                     
                       
                         
                           Σ 
                           
                             n 
                             = 
                             
                               - 
                               1 
                             
                           
                           0 
                         
                          
                         
                           Σ 
                           
                             l 
                             = 
                             0 
                           
                           1 
                         
                          
                         
                           cosh 
                            
                           
                             ( 
                             
                               2 
                                
                               ρ 
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   eal 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             ~ 
                                           
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               r 
                                               ~ 
                                             
                                             k 
                                           
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 j 
                                               
                                                
                                               
                                                 
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         
                                           - 
                                           j 
                                         
                                          
                                         
                                           
                                             l 
                                              
                                             
                                                 
                                             
                                              
                                             π 
                                           
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           Σ 
                           
                             n 
                             = 
                             1 
                           
                           2 
                         
                          
                         
                           Σ 
                           
                             l 
                             = 
                             0 
                           
                           1 
                         
                          
                         
                           cosh 
                            
                           
                             ( 
                             
                               2 
                                
                               ρ 
                                
                               
                                   
                               
                                
                               
                                 
                                   R 
                                   eal 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             ~ 
                                           
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               r 
                                               ~ 
                                             
                                             k 
                                           
                                            
                                           
                                              
                                             
                                               
                                                 - 
                                                 j 
                                               
                                                
                                               
                                                 
                                                   n 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   π 
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         
                                           - 
                                           j 
                                         
                                          
                                         
                                           
                                             l 
                                              
                                             
                                                 
                                             
                                              
                                             π 
                                           
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     26 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     for b k,2  and b k,1 , the second and first bit of the k-th DPSK symbol, respectively. 
     Alternatively or additionally, the high SNR approximation based on Equation (24) may be applied and the LLR in (26a) and (26b) may be respectively approximated by 
     
       
         
           
             
               
                 
                   
                     
                       LLR 
                       
                         k 
                         , 
                         2 
                       
                     
                     ≈ 
                     
                       2 
                        
                       ρ 
                        
                       
                         max 
                         
                           
                             
                               l 
                               = 
                               0 
                             
                             , 
                             1 
                           
                            
                           
                             
 
                           
                            
                           
                             
                               n 
                               = 
                               0 
                             
                             , 
                             1 
                           
                         
                       
                     
                   
                   | 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     M 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                               
                                 2 
                                  
                                 l 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   | 
                   
                     
                       - 
                       2 
                     
                      
                     ρ 
                      
                     
                       max 
                       
                         
                           
                             l 
                             = 
                             0 
                           
                           , 
                           1 
                         
                          
                         
                           
 
                         
                          
                         
                           
                             n 
                             = 
                             2 
                           
                           , 
                           3 
                         
                       
                     
                   
                   | 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     M 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                               
                                 2 
                                  
                                 l 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   | 
                 
               
               
                 
                   ( 
                   
                     27 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       LLR 
                       
                         k 
                         , 
                         1 
                       
                     
                     ≈ 
                     
                       2 
                        
                       ρ 
                        
                       
                         max 
                         
                           
                             
                               l 
                               = 
                               0 
                             
                             , 
                             1 
                           
                            
                           
                             
 
                           
                            
                           
                             
                               n 
                               = 
                               
                                 - 
                                 1 
                               
                             
                             , 
                             0 
                           
                         
                       
                     
                   
                   | 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     M 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                               
                                 2 
                                  
                                 l 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   | 
                   
                     
                       - 
                       2 
                     
                      
                     ρ 
                      
                     
                       max 
                       
                         
                           
                             l 
                             = 
                             0 
                           
                           , 
                           1 
                         
                          
                         
                           
 
                         
                          
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           2 
                         
                       
                     
                   
                   | 
                   
                     
                       R 
                       eal 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               
                                 r 
                                 ~ 
                               
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               
                                 
                                   r 
                                   ~ 
                                 
                                 k 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       n 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     M 
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                               
                                 2 
                                  
                                 l 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               M 
                             
                           
                         
                       
                       ] 
                     
                   
                   | 
                   . 
                 
               
               
                 
                   ( 
                   
                     27 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     C. Octal DPSK 
     In the case of Octal DPSK (ODPSK, M=8), when the example mapping rule of Table 3 is used, Equation (22) may be simplified as 
     
       
         
           
             
               
                 
                   
                     
                       LLR 
                       
                         k 
                         , 
                         3 
                       
                     
                     = 
                     
                       ln 
                        
                       
                         
                           
                             Σ 
                             
                               n 
                               = 
                               
                                 - 
                                 3 
                               
                             
                             0 
                           
                            
                           
                             Σ 
                             
                               l 
                               = 
                               0 
                             
                             3 
                           
                            
                           
                             cosh 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 ρ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     R 
                                     eal 
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               r 
                                               ~ 
                                             
                                             
                                               k 
                                               - 
                                               1 
                                             
                                           
                                           + 
                                           
                                             
                                               
                                                 r 
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     for b k,3 , b k,2  and b k,1 , respectively. The application of (24) to ODPSK for high SNR approximation may be straightforward. 
     In the case of M-DPSK, M&gt;8, LLR calculation may be straightforwardly derived based on the principles of Equations (22) and (24). Note that the techniques described in this disclosure may be applied to any M-DPSK mapping rules including those different than the example ones in Tables 1-4. The LLR calculation may be modified accordingly based on the mapping rules. 
     V. Simulation Results 
       FIG. 12  a plot  1200  illustrating example bit error rate (BER) performances of Binary-DPSK with (672,336)-LDPC under AWGN Channel. In performing the simulation, the used LDPC is specified by the IEEE 802.11 ad standard document. The plot  1200  includes the bit-error rates obtained via computer simulation using various LLR calculation techniques. As indicated, for a given SNR value, using Gaussian approximation (GA) according to Equation (15) for soft detection of BDPSK signal yields the highest BER compared to the other four illustrated techniques, although using Equation (17) or (18) shows negligible performance improvement over the case using GA in Equation (15). On the other hand, exploiting phase offset for soft detection of BDPSK signal (e.g., by estimating the phase offset θ and using Equation (25a) for LLR calculation based on the estimated phase offset {circumflex over (θ)}) demonstrate significant performance improvement (for example, about 0.9 dB SNR improvement at BER=10 −5 ) over the above three schemes. In the illustrated example, N=672 bits are used in the MLE for the phase offset estimation in simulation. Furthermore,  FIG. 12  shows that when Equation (25a) is used for LLR calculation, using the phase offset estimate (e.g., as obtained according to Equation (21)) yields the same (or substantially the same) error performance as that may be obtained by using the actual phase offset assuming the receiver knows the exact phase offset. 
       FIG. 13  is a plot  1300  illustrating example BER performances of Quaternary-DPSK and 8-DPSK with the same (672,336)-LDPC under AWGN Channel. In the illustrated example, N=336 symbols and N=896 symbols are used for phase offset estimation for QDPSK and 8-DPSK, respectively. As indicated, using the Equation (26) for soft detection of QDPSK signals provides an improvement of around 0.9 dB, while using Equation (28) for soft detection of 8-DPSK provides an improvement of about 0.5 dB over the GA approach based on Equation (15), respectively. 
     As demonstrated by computer simulation above, exploiting phase offset in soft detection of DPSK signal, for example, by estimating the phase offset and using the estimated phase offset in soft detection output (e.g., LLR) calculation may improve the error rate performance. In some instances, these techniques may help reduce the transmitter power without sacrificing the data rate and/or help increase data throughput (e.g., by increasing the order of modulation). 
       FIG. 14  is a flowchart illustrating an example method  1400  for soft-detection of M-ary DPSK signal. The receiver may be a receiver in a wireless or wireline communication system, or may be another appropriate device. The receiver may be, for example, the example receiver  104  in  FIG. 1 . In some instances, the receiver may be included in a network node (e.g., the example network node  200  described with respect to  FIG. 2 ) or a UE (e.g., the example UE  300  described with respect to  FIG. 3 ). In some implementations, the receiver may include identical or similar components to the example receiver  420  in  FIG. 4B  or have an identical or similar structure to one or more of the example receiver structure in  FIGS. 8-11 . In some implementations, the receiver may be configured in another manner. 
     At step  1410 , a DPSK signal is received at the receiver. The DPSK signal, for example, may include a sequence of DPSK modulated symbols. In some instances, the DPSK may be used in combination with a channel code (e.g., convolution code, turbo code, LDPC code, etc.). In some instances, the received M-ary DPSK signal may suffer from an unknown phase offset θ. The phase offset θ may be a random variable and be introduced, for example, by a channel that the M-ary DPSK signal passes through, by a phase drifting due to imperfect frequency synchronization, or any other possible factor. The phase offset θ may be constant for all received M-ary DPSK symbols (e.g., under an AWGN or flat fading channel), or invariant over a number of M-ary DPSK symbols (e.g., under a block-fading channel). 
     In some implementations, the DPSK may be M-ary DPSK, where M may be, for example, 2, 4, 8, 16, 32, 64, etc. In some implementations, the M-ary DPSK may optionally have a phase rotation φ in each symbol. For example, the φ-MDPSK. As an example, for φ=π/4, M==4, it is a typical π/4-QDPSK. 
     At step  1420 , the phase of the received φ-MDPSK signal is de-rotated. In some implementations, this step may be optional and it may only be performed for the φ-MDPSK signal. Specifically, the phase of the received signal is de-rotated by the same amount of the phase rotation φ introduced on the transmitter side. For example, the k-th received symbol may be de-rotated by kφ radians (i.e., rotated by −kφ radians). The resulting signal may be treated as a regular M-ary DPSK signal with φ=0. The phase of the received signal may be de-rotated prior to estimating the unknown phase offset of the received signal. 
     At step  1430 , the phase offset of the received DPSK signal is estimated or otherwise identified. For instance, the phase offset estimate may be denoted as {circumflex over (θ)}. In some implementations, the phase offset θ may be estimated based on the maximum likelihood (ML) principle, or any other appropriate principle. For example, the phase offset θ may be estimated according to the MLE described with respect to Equation (21). The example MLE, although it may produce a phase ambiguity, suffices to provide accurate phase offset estimation for soft detection of the DPSK signal as demonstrated in the simulation result presented in  FIG. 12 . 
     In some implementations, the phase offset estimation may be based solely on the received DPSK signal (e.g., according to the MLE in Equation (21)) without requiring dedicated training symbols. Alternatively or additionally, the phase offset θ may be estimated based on preambles, training symbols, or any other type of signal transmitted by the transmitter. As an example, the phase offset θ may be estimated based on preambles that are, for example, received prior to the DPSK signal. In one implementation, the receiver may estimate the phase offset θ based on the preamble and directly apply the estimated phase offset to soft detection of the DPSK signal (e.g., plugging the estimated phase offset into LLR calculation). In another example, the receiver may further perform a fine estimation based on the received DPSK signal and apply the fine-tuned phase offset estimate to calculating the soft detection output. In some implementations, the phase offset of the DPSK signal may be estimated or identified based on a phase offset estimation of another type of signal (e.g., PSK signal, QAM signal, etc.) that is transmitted in connection with the DPSK signal (e.g., within the same packet, sharing an identical or similar channel, etc.). Additional or different techniques may be used to estimate and identify the phase offset of the received DPSK signal. 
     At step  1440 , a soft detection output is calculated. The soft detection output may include a soft detection metric (e.g., log-likelihood ratio (LLR)) for each bit. In some implementations, the LLR of a single bit b k  may be derived based on the conditional joint probability density function (pdf) of two consecutively received DPSK symbols (e.g., r k  and r k-1 ) conditioned on the value of the b k  (e.g., b k =1 or b k =−1). The derivation of the soft detection output may explicitly account for and exploit the phase offset of the received DPSK signal. In some implementations, the receiver may employ the phase offset estimate {circumflex over (θ)} obtained at  1430  in LLR calculations. For example, the LLR for each bit of an M-ary DPSK symbol may be calculated according to Equation (22) or Equation (24) (e.g., when SNR is high). More specifically, the LLR for each bit of a BDPSK, QDPSK, and 8-DPSK symbol may be calculated according to Equations (25), (26)-(27), and (28), respectively. LLR for each bit of higher order of Mary QPSK (M&gt;8) and its respective high SNR approximation may be generalized based on the Equations (22) and (24) accordingly. In some implementations, each received M-ary DPSK symbol may be phased compensated based on the phase offset estimate {circumflex over (θ)} (for example, by subtracting the phase offset estimate {circumflex over (θ)} from the received signal, de-rotating the received signal by {circumflex over (θ)} radians, or rotating the received signal by −{circumflex over (θ)} radians) prior to calculating the soft detection metrics, for example, as shown in  FIG. 11 . 
     At step  1450 , the information bits are decoded and output, for example, by a channel decoder. The channel decoder may perform a decoding algorithm matched to the channel coding used by the transmitter. The channel decoder may include, for example, a turbo code decoder, an LDPC decoder, or any other type of decoder. The channel decoder may determine whether an output bit is 1 or −1 (or 0) based on the calculated soft detection output (e.g., the LLR for each bit). 
     In some instances, the receiver capable to exploit phase offset for soft detection of DPSK signal may be referred to as an advanced DPSK receiver. The advanced DPSK receiver may provide enhanced receiver performance compared to the legacy DPSK receiver that does not exploit phase offset for DPSK signals. In some implementations, in order to take advantage of receiver performance enhancement, the receiver may send a signaling to a transmitter to indicate that the receiver may provide enhanced receiver performance that includes, for example, exploiting the phase offset in soft detection of the DPSK signal. The signaling may be transmitted before, during, or after receiving the DPSK signal. As an example, the signaling may be transmitted prior to the transmitter transmitting DPSK modulated data. Based on such signaling, the transmitter may adjust its transmitting power, modulation order, channel coding scheme, or any other appropriate parameter to improve or optimize the system performance. In some implementations, the signaling may be one-bit control information that indicates whether the receiver is an advanced receiver or not. In some other implementations, the signaling may include more than one bit that may convey additional information that includes, for example, a quantitative performance improvement metric for M-ary DPSK signal, for M=2, 4, 8, 16 . . . . 
     VI. Exemplary Embodiments 
     In one embodiment, a method performed at a receiver of a communication system may be provided. The method may include receiving an Mary differential phase-shift keying (DPSK) signal containing a phase offset and optionally a phase rotation; estimating the phase offset of the received signal; and/or calculating a soft detection metric employing the estimated phase offset to provide enhanced receiver performance. The method may include subtracting the phase offset estimate from the received signal prior to calculating the soft detection metric and/or de-rotating the phase of the received signal by the same amount of the phase rotation prior to estimating the phase offset of the received signal. Estimating the phase offset of the received signal may be based on maximum likelihood principle. The method may include transmitting a signal or signaling, from the receiver to a transmitter, indicating that the receiver is capable of receiver performance enhancement. The soft detection metric may be a log-likelihood ratio (LLR) for soft detection of the received M-ary DPSK signal and the calculation of the LLR may be based upon a conditional joint probability density function of two consecutively received symbols based on equations of (22)-(28) noted above. The method may include additional, fewer, or alternative steps. 
     In another embodiment, a receiver of a communication network may be provided. The receiver may include one or more processors configured to: receive an M-ary differential phase-shift keying (DPSK) signal containing a phase offset and optionally a phase rotation; estimate the phase offset of the received signal; and/or calculate a soft detection metric employing the estimated phase offset to provide enhanced receiver performance. The one or more processors may further be configured to subtract the phase offset estimate from the received signal prior to calculating the soft detection metric and/or de-rotate the phase of the received signal by the same amount of the phase rotation prior to estimating the phase offset of the received signal. Estimating the phase offset of the received signal may be based on maximum likelihood principle. The one or more processors may further be configured to send a signal or signaling, from the receiver to a transmitter, indicating that the receiver is capable of receiver performance enhancement. The soft detection metric may be log-likelihood ratio (LLR) for soft detection of the received M-ary DPSK signal and the calculation of the LLR may be based on a conditional joint probability density function of two consecutively received symbols based on equations (22)-(28). The receiver may include additional, fewer, or alternate components and functionality. 
     In another embodiment, a method performed at a receiver of a communication system may be provided. The method may include receiving a signal including a sequence of differential phase-shift keying (DPSK) modulated symbols including an unknown phase offset; estimating the unknown phase offset of the received signal; and/or calculating a likelihood ratio for each bit of the DPSK modulated symbols based upon the estimated phase offset. The method may include subtracting the estimated phase offset from a phase of the received signal prior to calculating the likelihood ratio for each bit of the DPSK modulated symbols. The likelihood ratio may comprise a log-likelihood ratio (LLR) of each bit being 1 or −1. The likelihood ratio may be based on a conditional joint probability density function of two consecutively received symbols. The DPSK may be binary DPSK, quaternary DPSK, octal DPSK, or DPSK of higher orders. The method may further include de-rotating the received signal by a predefined phase prior to estimating the unknown phase offset of the received signal. Estimating the unknown phase offset of the received signal may be based on maximum likelihood principle. The method may include determining bit values for the DPSK modulated symbols using the calculated likelihood ratio for each bit. The method may include additional, fewer, or alternative steps. 
     In another embodiment, a receiver of a communication network may be provided. The receiver may include (1) means for receiving an M-ary differential phase-shift keying (DPSK) signal containing a phase offset and optionally a phase rotation; (2) means for estimating the phase offset of the received signal; and/or (3) means for calculating a soft detection metric employing the estimated phase offset to provide enhanced receiver performance. The receiver may include means for subtracting the phase offset estimate from the received signal prior to calculating the soft detection metric, and/or means for de-rotating the phase of the received signal by the same amount of the phase rotation prior to estimating the phase offset of the received signal. Estimating the phase offset of the received signal may be based on maximum likelihood principle. The “means for” functionality noted above may be provided by one or more processors and/or computer instructions stored on non-transitory memory. The receiver may include additional, fewer, or alternate components and functionality. 
     Although several illustrated examples are related to wireless communications, the example techniques described here may be applied to any other type of communications, such as wireline (e.g., cooper wire, fiber optical etc.) communication, satellite communication, etc. In addition, the example technique described in this disclosure may be applied to any other appropriate application that may involve signal modulation and detection (e.g., data storage, data compression, data recovery, etc.). 
     While several implementations have been provided in the present disclosure, it should be understood that the disclosed systems and methods may be embodied in many other specific forms without departing from the scope of the present disclosure. The present examples are to be considered as illustrative and not restrictive, and the intention is not to be limited to the details given herein. For example, the various elements or components may be combined or integrated in another system or certain features may be omitted, or not implemented. 
     Also, techniques, systems, subsystems and methods described and illustrated in the various implementations as discrete or separate may be combined or integrated with other systems, modules, techniques, or methods without departing from the scope of the present disclosure. Other items shown or discussed as coupled or directly coupled or communicating with each other may be indirectly coupled or communicating through some interface, device, or intermediate component whether electrically, mechanically, or otherwise. Other examples of changes, substitutions, and alterations are ascertainable by one skilled in the art and could be made without departing from the spirit and scope disclosed herein. 
     While the above detailed description has shown, described, and pointed out the fundamental novel features of the disclosure as applied to various implementations, it will be understood that various omissions and substitutions and changes in the form and details of the system illustrated may be made by those skilled in the art, without departing from the intent of the disclosure. In addition, the order of method steps are not implied by the order they appear in the claims.