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. ππΏ πππ€ππ πππππ’ππππ¦ πππππ‘ β 0 πππ ππ» π’ππππ πππππ’ππππ¦ πππππ‘ 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 βcentralβ frequency f o. ο· A Low pass signal is also a band-pass signal with lower frequency limit ππΏ = 0. Then, the upper frequency limit ππ» (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 π π = 0πππ π > π΅ Sampling Theorem for a Band limited signal is stated as: βA signal m(t) band limited to ππ π»π§ can be specified in terms of its samples taken for 1 every ππ β€ 2π π ππ, where ππ is referred to as Sampling interval. π The same can be expressed as ππ β₯ 2ππ π»π§ i.e. samples/sec where, ππ is the sampling frequency. ο Ideal Sampling(Instantaneous Sampling): ο 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. ο An ideally sampled signal may be regarded as the product of an analog signal x(t) and a periodic impulse train. Where π π‘ = β π=ββ πΏ( π‘ β πππ ) is the sampling signal which is a periodic impulse Train, with a period of ππ , which is referred to as Sampling period or Sampling interval. ο Sampled Signal π¦(t)= π₯(t).π (t) = π₯(t). β π=ββ π₯ π πΏ( π‘ β πππ ) β π=ββ πΏ (π‘ β πππ ) = β π=ββ π₯ πππ πΏ( π‘ β πππ ) =
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ο 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₯ π‘ . πΏ π‘ β π‘0 = π₯ π‘0 . πΏ π‘ β π‘0 ο The discrete signal x(n) represents the sequence of sample values π₯(πππ ) ο The above π¦ (t) in frequency domain is the convolution of the respective signals ο Periodic impulse train in time domain is also a periodic impulse train in frequency domain. ο Since, convolution with an impulse simply shifts a signal i.e{.π π β πΏ π β π0 = π π β π0 }, it follows that π π = β π=ββ π π β πππ . ο Thus, π π is a periodic function of π, consisting of shifted replicas of π(π) ο The trigonometric Fourier series representation of s(t) is given by β π π‘ = π0 + ππ . πΆππ πππ π‘ + ππ . πππ πππ π‘ π=1 1 Where π0 = π π ππ 2 ππ = 2 ππ ππ = 2 ππ π β π 2 ππ 2 ππ 2 π β π 2 1 s t . dt = π π s t . πΆππ πππ π‘ dt = π β π 2 ππ s t . πππ πππ π‘ dt = 0 2 Hence, 1 π π‘ = + ππ 1 Sampled Signal y(t)=x(t).s(t)= x(t)[π + π β π=1 β 2 π=1 π π 2 . πΆππ πππ π‘ ππ . πΆππ πππ π‘ ] 1 2 2 2 . π₯ π‘ + π₯ π‘ . πΆππ ππ π‘ + π₯ π‘ . πΆππ 2ππ π‘ + π₯ π‘ . πΆππ 3ππ π‘ + β β β ππ ππ ππ ππ Taking Fourier transform on both sides π¦ π‘ = π π = 1 [π π + π π β ππ + π π + ππ + π π β 2ππ + π π + 2ππ + β β β] ππ 1 =π π β π =ββ π π β πππ , π€ππππ π = 0, Β±1, Β±2, Β±3, β¦ ππ‘π.
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ο Thus, sampling in time domain results in periodicity in frequency domain. ο This is called ideal sampling or impulse sampling or Instantaneous sampling. ο Here, the sampling signal is a true impulse train. Reconstruction of Signal from its Sampled Version ο Consider the signal π¦ π‘ 1 2 2 . π₯ π‘ + π₯ π‘ . πΆππ ππ π‘ + π₯ π‘ . πΆππ 2ππ π‘ ππ ππ ππ 2 + π₯ π‘ . πΆππ 3ππ π‘ + β β β ππ ο The first term in the above π¦(t) is the baseband signalπ₯ π‘ itself. The remaining terms give rise to DSB-SC signals with carrier frequencies ππ , 2ππ , 3ππ β ππ‘π. whose spectra are symmetric about the respective carrier frequencies. ο The spectrum of π¦ (t) is same as original analog spectrum, but repeats at multiples of sampling frequency ππ . ο These higher order components which are centered on the multiples of ππ are referred to as image frequencies. = ο· Let the Sampling Frequency ππ < 2ππ , where ππ is the band limiting frequency of the signal π₯ π‘ . ο Various frequency components in the sampled signal are ππ , ππ β ππ (π€ππππ ππ πππ π π‘πππ ππ ), ππ + ππ , 2ππ Β± ππ , 3ππ Β± ππ ππ‘π. ο The image frequencies centered about ππ will fold over or alias into the baseband frequencies.
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