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Note for Signals and Stochastic Process - SSP By Chevella Anilkumar

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Signals and Stochastic Processes Module 12: Introduction to Sampling Objective: To Understand the concept of Sampling a signal its reconstruction and various methods of Sampling Module Introduction: In the applications of signal processing in real time systems, the mathematical description of signals will not be available. To find the characteristics of the signals, they must be measured and analyzed .If the signal is unknown, the process of analysis starts with the acquisition of the signal, which means measuring and recording the signals over a period of time.Sampling is the acquisition of a continuous signal at discrete time intervals. After Sampling, the analog signal is represented at discrete times only, with the values of the samples equal to those of the original analog signal at the discrete times. In the process of Analog to Digital conversion of a signal, the signal is first sampled, converting a continuous ,analog signal into a discrete time, continuous amplitude signal. Next comes the process of Quantization and digitization. The present Module focuses on the methods of sampling a given low pass signal and its reconstruction from its sampled version. Module Description: ο‚· A Band-pass signal is a signal containing a band of frequencies i.e. its magnitude spectrum ranges over two frequency limits i.e. 𝑓𝐿 π‘™π‘œπ‘€π‘’π‘Ÿ π‘“π‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ π‘™π‘–π‘šπ‘–π‘‘ β‰  0 π‘Žπ‘›π‘‘ 𝑓𝐻 π‘’π‘π‘π‘’π‘Ÿ π‘“π‘Ÿπ‘’π‘žπ‘’π‘’π‘›π‘π‘¦ π‘™π‘–π‘šπ‘–π‘‘ β‰  0 . ο‚· A Band-pass Signal is a signal x ( t) whose Fourier transform X( f ) is nonzero only in some small band around some β€œcentral” frequency f o. ο‚· A Low pass signal is also a band-pass signal with lower frequency limit 𝑓𝐿 = 0. Then, the upper frequency limit 𝑓𝐻 (β‰  0) is referred to as Band-limiting frequency of the signal, and the Low pass signal is also referred to as band-limited signal.

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ο‚· A signal x(t) band-limited to B Hz will have its Fourier transform 𝑋 𝑓 = 0π‘“π‘œπ‘Ÿ 𝑓 > 𝐡 Sampling Theorem for a Band limited signal is stated as: β€œA signal m(t) band limited to π‘“π‘š 𝐻𝑧 can be specified in terms of its samples taken for 1 every 𝑇𝑠 ≀ 2𝑓 𝑠𝑒𝑐, where 𝑇𝑠 is referred to as Sampling interval. π‘š The same can be expressed as 𝑓𝑠 β‰₯ 2π‘“π‘š 𝐻𝑧 i.e. samples/sec where, 𝑓𝑠 is the sampling frequency. οƒ˜ Ideal Sampling(Instantaneous Sampling): οƒ˜ Ideal Sampling describes a sampled signal as a weighted sum of impulses, with weights being equal to the values of the analog signal at the location of the impulses. οƒ˜ An ideally sampled signal may be regarded as the product of an analog signal x(t) and a periodic impulse train. Where 𝑠 𝑑 = ∞ 𝑛=βˆ’βˆž 𝛿( 𝑑 βˆ’ 𝑛𝑇𝑠 ) is the sampling signal which is a periodic impulse Train, with a period of 𝑇𝑠 , which is referred to as Sampling period or Sampling interval. οƒ˜ Sampled Signal 𝑦(t)= π‘₯(t).𝑠(t) = π‘₯(t). ∞ 𝑛=βˆ’βˆž π‘₯ 𝑛 𝛿( 𝑑 βˆ’ 𝑛𝑇𝑠 ) ∞ 𝑛=βˆ’βˆž 𝛿 (𝑑 βˆ’ 𝑛𝑇𝑠 ) = ∞ 𝑛=βˆ’βˆž π‘₯ 𝑛𝑇𝑠 𝛿( 𝑑 βˆ’ 𝑛𝑇𝑠 ) =

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οƒ˜ Because of the sampling property of the unit impulse , multiplying x(t) by a unit impulse samples the value of the signal at the point at which the impulse is located i.e.π‘₯ 𝑑 . 𝛿 𝑑 βˆ’ 𝑑0 = π‘₯ 𝑑0 . 𝛿 𝑑 βˆ’ 𝑑0 οƒ˜ The discrete signal x(n) represents the sequence of sample values π‘₯(𝑛𝑇𝑠 ) οƒ˜ The above 𝑦 (t) in frequency domain is the convolution of the respective signals οƒ˜ Periodic impulse train in time domain is also a periodic impulse train in frequency domain. οƒ˜ Since, convolution with an impulse simply shifts a signal i.e{.𝑃 πœ” βˆ— 𝛿 πœ” βˆ’ πœ”0 = 𝑃 πœ” βˆ’ πœ”0 }, it follows that π‘Œ πœ” = ∞ 𝑛=βˆ’βˆž 𝑋 πœ” βˆ’ π‘›πœ”π‘  . οƒ˜ Thus, π‘Œ πœ” is a periodic function of πœ”, consisting of shifted replicas of 𝑋(πœ”) οƒ˜ The trigonometric Fourier series representation of s(t) is given by ∞ 𝑠 𝑑 = π‘Ž0 + π‘Žπ‘› . πΆπ‘œπ‘  π‘›πœ”π‘  𝑑 + π‘Žπ‘› . 𝑆𝑖𝑛 π‘›πœ”π‘  𝑑 𝑛=1 1 Where π‘Ž0 = 𝑇 𝑠 𝑇𝑠 2 π‘Žπ‘› = 2 𝑇𝑠 𝑏𝑛 = 2 𝑇𝑠 𝑇 βˆ’ 𝑠 2 𝑇𝑠 2 𝑇𝑠 2 𝑇 βˆ’ 𝑠 2 1 s t . dt = 𝑇 𝑠 s t . πΆπ‘œπ‘  π‘›πœ”π‘  𝑑 dt = 𝑇 βˆ’ 𝑠 2 𝑇𝑠 s t . 𝑆𝑖𝑛 π‘›πœ”π‘  𝑑 dt = 0 2 Hence, 1 𝑠 𝑑 = + 𝑇𝑠 1 Sampled Signal y(t)=x(t).s(t)= x(t)[𝑇 + 𝑠 ∞ 𝑛=1 ∞ 2 𝑛=1 𝑇 𝑠 2 . πΆπ‘œπ‘  π‘›πœ”π‘  𝑑 𝑇𝑠 . πΆπ‘œπ‘  π‘›πœ”π‘  𝑑 ] 1 2 2 2 . π‘₯ 𝑑 + π‘₯ 𝑑 . πΆπ‘œπ‘  πœ”π‘  𝑑 + π‘₯ 𝑑 . πΆπ‘œπ‘  2πœ”π‘  𝑑 + π‘₯ 𝑑 . πΆπ‘œπ‘  3πœ”π‘  𝑑 + βˆ’ βˆ’ βˆ’ 𝑇𝑠 𝑇𝑠 𝑇𝑠 𝑇𝑠 Taking Fourier transform on both sides 𝑦 𝑑 = π‘Œ πœ” = 1 [𝑋 πœ” + 𝑋 πœ” βˆ’ πœ”π‘  + 𝑋 πœ” + πœ”π‘  + 𝑋 πœ” βˆ’ 2πœ”π‘  + 𝑋 πœ” + 2πœ”π‘  + βˆ’ βˆ’ βˆ’] 𝑇𝑠 1 =𝑇 𝑠 ∞ 𝑛 =βˆ’βˆž 𝑋 πœ” βˆ’ π‘›πœ”π‘  , π‘€π‘•π‘’π‘Ÿπ‘’ 𝑛 = 0, Β±1, Β±2, Β±3, … 𝑒𝑑𝑐.

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οƒ˜ Thus, sampling in time domain results in periodicity in frequency domain. οƒ˜ This is called ideal sampling or impulse sampling or Instantaneous sampling. οƒ˜ Here, the sampling signal is a true impulse train. Reconstruction of Signal from its Sampled Version οƒ˜ Consider the signal 𝑦 𝑑 1 2 2 . π‘₯ 𝑑 + π‘₯ 𝑑 . πΆπ‘œπ‘  πœ”π‘  𝑑 + π‘₯ 𝑑 . πΆπ‘œπ‘  2πœ”π‘  𝑑 𝑇𝑠 𝑇𝑠 𝑇𝑠 2 + π‘₯ 𝑑 . πΆπ‘œπ‘  3πœ”π‘  𝑑 + βˆ’ βˆ’ βˆ’ 𝑇𝑠 οƒ˜ The first term in the above 𝑦(t) is the baseband signalπ‘₯ 𝑑 itself. The remaining terms give rise to DSB-SC signals with carrier frequencies 𝑓𝑠 , 2𝑓𝑠 , 3𝑓𝑠 βˆ’ 𝑒𝑑𝑐. whose spectra are symmetric about the respective carrier frequencies. οƒ˜ The spectrum of 𝑦 (t) is same as original analog spectrum, but repeats at multiples of sampling frequency 𝑓𝑠 . οƒ˜ These higher order components which are centered on the multiples of 𝑓𝑠 are referred to as image frequencies. = ο‚· Let the Sampling Frequency 𝑓𝑠 < 2π‘“π‘š , where π‘“π‘š is the band limiting frequency of the signal π‘₯ 𝑑 . οƒ˜ Various frequency components in the sampled signal are π‘“π‘š , 𝑓𝑠 βˆ’ π‘“π‘š (𝑀𝑕𝑖𝑐𝑕 𝑖𝑠 𝑙𝑒𝑠𝑠 π‘‘π‘•π‘Žπ‘› π‘“π‘š ), 𝑓𝑠 + π‘“π‘š , 2𝑓𝑠 Β± π‘“π‘š , 3𝑓𝑠 Β± π‘“π‘š 𝑒𝑑𝑐. οƒ˜ The image frequencies centered about 𝑓𝑠 will fold over or alias into the baseband frequencies.

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