EE 422G - Signals and Systems LaboratoryLab 7 Bit Detection with the Correlation ReceiverKevin D. DonohueDepartment of Electrical and Computer EngineeringUniversity of KentuckyLexington, KY 40506February 27, 2013EE 422G - Signals and Systems LaboratoryLab 7 Bit Detection with the Correlation ReceiverKevin D. DonohueDepartment of Electrical and Computer EngineeringUniversity of KentuckyLexington, KY 40506February 27, 2013Objectives:- Know the basic operation of the optimal matched filter and correlation receivers for detecting known/deterministic waveforms in noise.- Implement the correlation receiver for detecting signals in noise.- Implement the match filter for decoding bit streams received from noisy, bandlimited channels, and computing bit error rates.1. Background:In modern digital communication systems information from signals, such as speech,video, and text must be transferred to other locations. Noise corrupts the signal duringtransmission, which arises from many sources, including interference from othertransmitters and thermal motion of the electrons. The purpose of the receiver is toaccurately recover the transmitted information, even in the presence of noise anddistortion.The problem of recovering signals corrupted by noise has several solutions. Thecorrelation receiver can be shown to be the best solution (optimal) for recovering aknown waveform in the presence of additive white Gaussian noise (AWGN). Acorrelation receiver basically compares the received signal to a set of waveforms itexpects to receive. This applies directly to a digital communication problem since eachbit (or bit sequence) can be encoded to a set of distinct waveforms known to both thetransmitter and receiver. For example, consider the signal of Fig. 1a. If the transmittedwaveform is corrupted by noise, the received signal may look like the solid-lined plots ofFigs. 1b and c. A correlation operation aligns the received signals values with a template(shown in the dashed lines of Figs. 1b and c), and the sum of products between thereceived signal and template are taken over the duration of the waveform template. Thecorrelation will be greatest when the template is aligned with the received signal, asshown in Fig. 1b. A lesser response occurs as the misalignment increases. Fig. 1c showsa misalignment that results in the minimum response (largest negative number). Thecorrelation receiver often uses signals that are zero-mean to exploit the cancellationbetween positive and negative values in the correlation sum.(a)(b)(c)Figure 1. (a) Ideal transmitted waveform. (b) Matched alignment with noisy received waveform for maximum correlation receiver response. (c) Mismatched alignment with noisy received waveform for minimum correlation receiver response. A correlation receiver therefore parses the signal into intervals that are synchronized with the bit intervals, and integrates the product of the received signal and template over the interval to produce correlation values. The correlation values become detection statistics used to decide on the most likely bit value that was transmitted for each interval.In the case of binary signaling (0 or 1) there are only 2 possible symbols, so 2 correlation receivers would operate in parallel corresponding to waveforms associated with each bit. The bit channel with the greatest correlation value corresponds to the bit detection/decision. The synchronization of the segment intervals requires that both the source and receiver have the same clock signal (or an alternative way to synchronize) before applying the correlation receiver, since the alignment over the correlation interval affects the result. An example of this receiver is shown in Fig 2, where f0(t ) is the template for the 0 bit,f1(t ) is the template for the bit 1, and T is the synchronized bit interval. Every T seconds the input is correlated with the template and the correlation output is sampled to obtain the detection statistics. In this case the decision rule is to simply decide on the bit whose template has the best match (largest value). A bit error occurs when noise or bandwidth limitations result in an incorrect decision. For a given test or experiment, the numbers of errors per bit is referred to as the bit error rate (BER), and the expected value of the BER is the probability of error.0,1,1,0,0,1,0⋯¿¿¿Figure 2. Correlation receiver synchronized to waveform intervals for bit sequence detection.For applications where a physical event emits signals apart from a clock synchronization (such as a blood pressure drop, or an echo from an active sonar), the receiver must detect the signals as well as estimate when the signal was received. This is typically the case forsonar or radar systems, where the time of the received echo depends on the distance between the target and receiver (which can vary as the target moves through the space of interest). The detection of the echo signal indicates the presence of a target, and the time at which the echo returns indicates the distance between the target and receiver. In this case, intervals for correlation cannot be predetermined. Therefore, the signal template is slid continuously over the received signal while applying the correlation operation with the template. This produces a continuous output from which decisions are made on whether a target is present. This is typically done with a simple threshold. If there is a match with the signal of interest at a particular alignment, the output will exceed typical values corresponding to the no-signal case. An example of the correlation filter implemented as a matched filter for this case is shown in Fig. 3. The signal of interest is atapered sine wave. The top set of 3 waveforms in Fig. 3 show the signal, added noise, and the continuous sequence of detection statistics for the template that matches the signal. The lower 3 waveforms show the case when no signal is present (noise only). Note that detection statistics near 0.2 seconds reach a maximum for the case when a signal is present. This value is at least one order of magnitude greater than any of the outputs for the noise only signal. If accurate statistics of the noise fluctuations are available, the threshold can be set to achieve a specific false-alarm rate.