Telecom Tutorials by Samir Amberkar

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INDEX Background » 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)

»	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 harmonically related continuous-time sinusoids. F0 is fundamental frequency. Same combination for discrete-time sinusoids is shown below. Note that in case of discrete-time sinusoid, frequency is limited as seen earlier. So, k takes values 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 sampling 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

»	Discrete-Time systems

»	Types of discrete-time systems Static (memoryless) Dynamic (with memory) Time invariant Time variant Relaxed or Not releaxed 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

»	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 for a certain input signal, we can calculate response of the system for the same input signal. 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 zero, system has non-zero initial condition and it is *not* releaxed. 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.

»	Example of a simple recursive system 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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