Telecom Tutorials by Samir Amberkar

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  "urge the reader to use/trust the content only after verifying it against standards and/or consulting it with experts in the field".

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INDEX Background » Books » Numbers » Complex number » Euler's formula » IQ Modulation and Demodulation Digital Signal Processing » Signal processing » Digital Signal Processing system » Continuous-Time and Discrete-Time Sinusoid signals » Discrete-Time Sinusoid signal » Complex Sinusoid (or Exponential) » Harmonically related Complex Sinusoids » Sampling of a sinusoid » Aliasing » Sampling theorem » Quantisation Discrete-Time Signals and Systems » Elementary discrete-time signals » Energy of discrete-time signal » Power of discrete-time signal » Simple operations on discrete-time signals » Discrete-Time systems » Types of discrete-time systems » Cascade (Serial) interconnection » Parallel interconnection » Linearity and Time invariance » Splitting discrete time signal in unit impulses » Convolution sum for discrete-time system » Commutativity of Convolution sum » Associativity of Convolution sum » Cascade connection and Convolution sum » Distributive law of Convolution sum » Check if system is Causal » Convolution sum for Causal systems » Check if system is Stable » Response of stable system to finite duration input signal » Duration of unit impulse response » Recursive system » Constant coefficient difference equation » Example of a simple recursive system » Solution of difference equation (direct method) » Direct Form I and II realisations » Correlation of Discrete-Time signals » Correlation as a Convolution » Autocorrelation » Crosscorrelation between finite duration discrete-time signals » Autocorrelation and Energy of discrete-time signals » Crosscorrelation and Energy of discrete-time signals » Correlation of discrete-time power signals » Autocorrelation to find periodicity » Collated observations and principals

»	Books Below book if you want to start studying DSP fundamentals, Digital Signal Processing by Proakis and Manolakis Below book if you are aiming for DSP in the context of Communication (say 5G NR), Digital Signal Processing in Modern Communication Systems by Schwarzinger

»	Numbers Below is our number scale. Complex numbers is superset of all numbers.

»	Complex number Rectangular format Polar format Relation between formats Related article: Quadratic polynomial with Complex roots.

»	Euler's formula

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IQ Modulation and Demodulation In Modulation, Amplitude and Phase of the Carrier is adjusted. This can be achieved with an "in-phase" component and an "quadrature" component as shown mathematically below. f is Carrier frequency. Demodulation (i.e. getting back "in-phase" and "quadrature" components) is achieved by low pass filter (LPF).

»	Signal processing Example,

»	Digital Signal Processing system

»	Continuous-Time and Discrete-Time Sinusoid signals Continuous-Time Discrete-Time

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Discrete-Time Sinusoid signal Fundamental difference between continuous-time and discrete-time sinusoid signals is that maximum rate of oscillation for discrete-time sinusoid signal is π ! In other words, discrete-time sinusoid signals with oscillation rate separated by 2π are identical. Taking cos(ωn) as discrete-time sinusoid, below is plot of it for increasing values of ω. Maximum oscillation frequency reached at ω = π. Oscillation frequency start decreasing till it reaches zero. As ω = 2πf, we may say that discrete-time sinusoid signals with frequency difference of 1 are identical. In other words, maximum value of f is 1/2. Usually, range of ω between −π and π is taken for analysis. Corresponding range of f will be between −1/2 and 1/2.

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Complex Sinusoid (or Exponential) Complex Sinusoid (or Exponential) is a function of frequency and time as shown below. If values (complex numbers) are plotted on rectangular coordinates, it will look like a point revolving in circular motion with frequency f. Anticlockwise if forward in time, clockwise otherwise.

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Harmonically related Complex Sinusoids Two sinusoids are said to be harmonically related if their frequencies are multiple of single frequency. This frequency is known as fundamental frequency. Below is a linear combination of harmonically related continuous-time sinusoids. F0 is fundamental frequency. Same combination for discrete-time sinusoids is shown below. In case of discrete-time sinusoid, as seen earlier, waveforms are same when oscillation rate is separated by 2π. If N is oscillation period (corresponding to F0), waveforms with k=0 and k=N will be same. That means, it will be sufficient if we take k from 0 to N-1.

»	Sampling of a sinusoid Analog to Digital conversion requires Sampling of the analog signal. Sampling is usually periodic. Below diagram shows sampling of a sinusoid with two frequencies, but with the same sampling period.

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Aliasing In Sampling illustration, though frequency of input sinusoid was increased, sampling period was kept same. If we try the experiment of increasing frequency of input sinusoid without reducing sampling period, we will notice that waveform based on sampled values start looking different from input signal. Later at some point, sampled values will show completely different sinusoid. In below diagram input signals have different frequencies, but sampled values are same. This is known as Aliasing. If we relate sampling with discrete-time, we can say that the maximum frequency of the sampled waveform is half of sampling frequency. The waveform (or sampled values) will repeat after crossing half of sampling frequency. Half of sampling frequency is known as folding frequency. It also means that if we know frequency of input signal (say F0), we can figure out another frequency (say F1) which will have identical sampled values for a given sampling frequency (say Fs).

»	Sampling theorem Below is an equation to get back input analog signal from sampled values.

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Quantisation During sampling, each sample value of continous-amplitude signal is expressed into certain number of digits (usually bits). The process is called Quantisation. Quantisation introduces a certain error due the conversion of continous-value to discrete-value; this error is known as Quantisation error. Information loss due to Quantisation error could be measured in terms of Signal-to-Quantisation noise ratio (SQNR). Each bit is equivalant to 6 dB power for sinusoidal signal as shown below.

»	Elementary discrete-time signals Unit impulse Unit step Energy signal Power signal Periodic signal Nonperiodic or Aperiodic signal Signal which is *not* periodic Symmetric (even) signal Antisymmetric (odd) signal

»	Energy of discrete-time signal Below is a definition of Energy and an example calculation for Unit impulse signal. As Energy of Unit impulse signal is finite, it is known as an Energy signal.

»	Power of discrete-time signal Below is a definition of Power and an example calculation for Unit step signal. As Power of Unit step signal is finite, it is known as Power signal.

»	Simple operations on discrete-time signals Shifting Folding Down-sampling (time scaling) Amplitude scaling Below diagrams illustrates various operations.

»	Discrete-Time systems

»	Types of discrete-time systems Static (memoryless) Dynamic (with memory) Time invariant Time variant Relaxed or Not relaxed Linear Nonlinear Does *not* satisfy above linear equation. Causal or Noncausal Recursive or Nonrecursive Stable Unstable

»	Cascade (Serial) interconnection

»	Parallel interconnection

»	Linearity and Time invariance Linearity should be not be related to Time invariance. Linearity deal with Amplitude or value (y axis) of the signal whereas Time invariance deal with Time (x axis).

»	Splitting discrete time signal in unit impulses

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Convolution sum for discrete-time system Convolution sum implies that if we know unit impulse response of the linear, relaxed, and time invariant discrete system, we can calculate response of the system for arbitrary input signal by convolution sum of the input signal with unit impulse response of the system. Below is an example of Convolution sum.

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Commutativity of Convolution sum In other words, if we excite a linear time invariant discrete-time system - which has unit impulse response as Tδ - with Finput signal, we get same response that we will receive when we excite another system - which has unit impulse response as Finput - with Tδ signal.

»	Associativity of Convolution sum

»	Cascade connection and Convolution sum

»	Distributive law of Convolution sum

»	Check if system is Causal

»	Convolution sum for Causal systems

»	Check if system is Stable

»	Response of stable system to finite duration input signal

»	Duration of unit impulse response

»	Recursive system

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Constant coefficient difference equation First term indicates that the system has a "State". If the input is zero, first term is revealed. So, it is known as Zero-Input Response (or Natural response). Let us call it FZI. It is obvious that if zero-input response is non-zero, system has non-zero initial condition and it is *not* relaxed. Second term is revealed when FZI = 0 i.e. when the system has no state or zero state. So, the second term is known as Zero-State Response. Let us call it FZS. So, It can be shown be that recursive system decribed by a linear constant-coefficient difference equation is linear and time invariant.

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Example of a simple recursive system Above equation indicates that FZS is convolution sum of Finput(n) and Anu(n) (refer Convolution sum for Causal systems). If the system is relaxed (i.e. FZI(n) = 0), FZS is same as the response of the system. Combining above two points, we can say that the impulse reponse of above first-order recursive and relaxed system is Anu(n). Another important point is: originally recursive, the system has been converted to non-recursive system ! The reverse should then be possible too. In general, every FIR LTI system can be realised both recursively and non-recursively. Let us check the stability of the system.

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Solution of difference equation (direct method) To find a solution to linear constant coefficient difference equation is to find an explicit equation for Foutput in terms of its initial conditions for a particular input. Below is a direct method. Find homegeneous or complementary solution. Example of finding homegeneous or complementary solution. Now let us look at particular solution. Example of finding particular solution. Let us check full solution for our example. Let us see if we get same solution by manual calculation. Solutions from Manual calculation and that from Direct method match ! Another point to note here is that the particular solution can be obtained from zero-state response by below equation. Taking our example. This matches with earlier calculated particular solution. As particular solution is found by making n approach infinity, it is called steady-state response. The portion of the solution that dies out is called transient response.

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Direct Form I and II realisations Direct Form I is one way of realisation of LTI system described by linear constant coefficient difference equation. The same can be converted to Form II. Both forms will require same number of multiplcations (M + N + 1), but Form II will require lesser number of delay elements. Form I will require (M + N) delays whereas Form II will require max{M,N} delays.

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Correlation of Discrete-Time Energy signals As put in book Digital Signal Processing by Proakis and Manolakis, "A mathematical operation that closely resembles Convolution is Correlation." In Telecommunication, this concept could be applied to check whether (part of) the received signal match to reference signal (pattern) for 0 or 1. Below equation describes Crosscorrelation between two discrete-time finite energy signals. Let us check whether order of the signals matter in Crosscorrelation.

»	Crosscorrelation as a Convolution Convolution and Crosscorrelation equations looks similar, here is how Crosscorrelation can be computed as a Convolution.

»	Autocorrelation Autocorrelation is simply crosscorelation with itself, as shown below. Here is another observation about Autocorrelation.

»	Crosscorrelation between finite duration discrete-time energy signals In case of finite duration discrete-time energy signals, Crosscorrelation can be expressed as limited summation.

»	Autocorrelation and Energy of discrete-time energy signals There is definite relation between Autocorrelation and Energy of discrete-time signals as shown below.

»	Crosscorrelation and Energy of discrete-time energy signals Let us took at relation between Crosscorrelation and Energy of discrete-time signals. Below examples illustrates above point.

»	Correlation of discrete-time power signals

»	Autocorrelation to find periodicity Below examples illustrates above point. We can see distinct periodic peaks, matching the periodicity of transmitted signal.

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Collated observations and principals Fundamental difference between continuous-time and discrete-time sinusoid signals is that maximum rate of oscillation for discrete-time sinusoid signal is π ! In other words, discrete-time sinusoid signals with oscillation rate separated by 2π are identical. More info here. If we know frequency of input signal, we can figure out another frequency which will have identical sampled values for a given sampling frequency. More info here. Convolution sum implies that if we know unit impulse response of the linear, relaxed, and time invariant discrete system, we can calculate response of the system for arbitrary input signal by convolution sum of the input signal with unit impulse response of the system. More info here. If we excite a linear time invariant discrete-time system - which has unit impulse response as Tδ - with Finput signal, we get same response that we will receive when we excite another system - which has unit impulse response as Finput - with Tδ signal. More info here. If two DT LTI systems interconnected in cascade fashion, their combined unit impulse response is convolution sum of their individual unit impulse responses. More info here. If two DT LTI systems are connected in cascade fashion, their order can be interchanged. More info here. If two DT LTI systems are connected in parallel fashion, their combined unit impulse response sum of their individual unit impulse responses. More info here. DT LTI system is causal only if its unit impulse response is zero for negative values of n. More info here. DT LTI system is stable only if sum of its unit impulse responses is bounded. More info here. When stable system is excited with finite duration input, its output dies out eventually. More info here. FIR systems are said to have finite memory whereas IIR systems are said to have infinite memory. More info here. Recursive system decribed by a linear constant-coefficient difference equation is linear and time invariant. More info here. For a linear and time invariant recursive system to be stable, only a fraction of output must be taken in feedback. More info here. Every FIR LTI system can be realised both recursively and non-recursively. More info here. Direct Form II realisation requires lesser number of delay elements than Direct Form I realisation. More info here. As put in book Digital Signal Processing by Proakis and Manolakis, "A mathematical operation that closely resembles Convolution is Correlation." More info here. Autocorrelation of Energy signal reaches its maximum value at n=0 and it is nothing but the energy of the signal. More info here and here. Correlation (both Crosscorrelation and Autocorrelation) are related to energy of Energy signal (and so power of the periodic signal). More info here. By Autocorrelation, we can find the periodicity of the periodic (power) signal mixed with random signal (say white noise). More info here.

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