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lecture8 - ECE 5670 Digital Communications Lecture 8 Pulse...

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ECE 5670 : Digital Communications Lecture 8: Pulse Shaping and Sampling 1 2/17/2011 Instructor: Salman Avestimehr Introduction Information is digital in today’s world but the physical world is still analog. Digital commu- nication entails mapping digital information into electromagnetic energy (voltage waveforms) and transmitting over an appropriate physical medium (over a wire or wireless). At the re- ceiver, we record the electromagnetic energy (voltage waveform again) and based on this knowledge, try to recover the original information bits. In the first lecture, we pointed out that for engineering convenience, the mapping between digital information and analog voltage waveforms is divided into two separate parts. At the transmitter: we first map digital information into a discrete sequence of voltage levels; this is the modulation or coding step. next, we interpolate between these voltage levels to produce an analog voltage waveform that is then transmitted; this is the DAC (digital to analog conversion) step. At the receiver: we sample the received analog voltage waveform to produce a discrete sequence of voltage levels; this is the ADC (analog to digital conversion) step. next, we map the discrete sequence of sampled voltage levels to the information bits; this is the demodulation or decoding step. These operations are depicted in Figure 1, in the context of transmission over the AWGN channel. We have seen in the previous lectures, in substantial detail, the steps of modulation (coding) and demodulation (decoding). In this lecture, we will delve deeper into the DAC and ADC steps. At the end of this lecture, we will be able to derive a relationship between the sampling and interpolation rates (of the ADC and DAC, respectively) with an important physical parameter of an analog voltage waveform: bandwidth . Digital to Analog Conversion (DAC) How do we map a sequence of voltages, { x [ m ] } , to waveforms, x ( t )? This mapping is known as DAC. There are a few natural conditions we would like such a mapping to meet: 1 Based on lecture notes of Professor Pramod Viswanath at UIUC. 1
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ˆ bits A/D D/A Demodulation (ML Rule) Modulation a45 a45 a27 a27 a63 a27 a45 x ( t ) y ( t ) x [ m ] y [ m ] bits w ( t ) a108 Figure 1: The basic transmit and receive operations in the context of communicating over the AWGN channel. 1. Since the digital information is contained in the discrete sequence of voltages { x [ m ] } , we would like these voltages to be readily recovered from the voltage waveform x ( t ). One way of achieving this is to set x ( mT ) = x [ m ] (1) where T is the time period between voltage samples. This way, all the information we want to communicate is present in the transmitted waveform and readily extractable too. 2. We could potentially pick any waveform that satisfies Equation (1). In other words, we seek to interpolate between the uniformly spaced voltage sequence. Of course, this interpolation should be universal, i.e., it should work for any sequence of discrete volt- age levels (this is because the voltage sequence varies based on the coding method (sequential vs block) and also the information bits themselves). While such interpola-
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