L5 - L5 Sampling reconstruction and software radio Until...

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L5 Sampling, reconstruction, and software radio Until this point, your study of signals and systems has concerned only the continuous-time case 1 , which dominated the early history of signal processing. About 50 years ago, however, the devel- opment of the modern computer generated research interest in digital signal processing (DSP), a type of discrete-time signal processing. Although hardware limitations made most real-time DSP impractical at the time, the continuing maturation of the computer has been matched with a continuing expansion of DSP. Much of that expansion has been into areas previously dominated by continuous-time systems: our telephone network, medical imaging, music recordings, wireless communications, and many more. You do not need to worry whether the time and e f ort you have invested in studying continuous- time systems will be wasted because of the growth of DSP — digital systems are practically always hybrids of analog and digital sub-systems. Furthermore, many DSP systems are linear and time- invariant, meaning that the same analysis techniques apply, although with some modiﬁcations. In this lab, you will explore some of the parallels between continuous-time systems and DSP with a “software radio” designed to the same speciﬁcations as the receiver circuit you developed on your protoboard. 1 Prelab Our software radio is typical of many DSP systems in that both the available input and re- quired output are continuous-time signals. The conversion of a continuous-time input signal to a discrete-time signal is called sampling (or A/D conversion), and the conversion of a discrete- time signal to a continuous-time output signal is called reconstruction (or D/A conversion). As discussed in class, samples f ( nT ) of a bandlimited analog signal f ( t ) can be used to re- construct f ( t ) exactly when the sampling interval T and signal bandwidth Ω = 2 π B satisfy the Nyquist criterion T < 1 2 B . This is illustrated by the hypothetical system shown in Figure L5.1, where the analog signal f ( t ) deﬁned at the output stage of a low-pass ﬁlter H 1 ( ω ) has a bandwidth Ω = 2 π B limited by the bandwidth Ω 1 = 2 π B 1 of the ﬁlter. A/D converter extracts the samples f ( nT ) from f ( t ) with a sampling interval of T , and D/A conversion of samples f ( nT ) into an analog signal y ( t ) can be envisioned as low-pass ﬁltering of a hypothetical signal f T ( t ) = n f ( nT ) δ ( t - nT ) using the ﬁlter H 2 ( ω ) . With an appropriate choice of H 2 ( ω ) , system output y ( t ) will be identical to f ( t ) in all its details as long as T < 1 2 B 1 . The reason for that can be easily appreciated after comparing the Fourier transforms F ( ω ) and F T ( ω ) of signals f ( t ) and f T ( t ) with the help of Figure L5.2. 1

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L5 - L5 Sampling reconstruction and software radio Until...

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