Discrete -time signals➔Discrete-time signals are signals which are defined only at discrete instants Discrete – Time System

Discrete – Time System Properties➔1. Linearity:- A system is said to be linear if it satisfies

of time. For those signals, the amplitude between the two time instants is just not defined. For Exponential sequence➔ It is defined as 𝑥[𝑛] = 𝑎n 𝑓𝑜𝑟 𝑎𝑙𝑙 n//a. If a is real, x[n] is a real exponential. superposition principle, which in turn is a combination of additivity and homogeneity. Additivity
discrete- time signal the independent variable is time n, and it is represented by x (n). A discrete time implies that If the response of the DT system to x1[n] is y1[n], and the response to x2[n] is y2[n], then
signal is not defined at instants between two successive samples. the response of the system to {x1[n]+x2[n]} must be {y1[n]+y2[n]}. Homogeneity implies that if the
Representing discrete-time signals➔Graphical Representation➔ Consider a single x (n) with response of a DT system to x[n] is y[n], then the response of the system to ax[n] must be ay[n], where
values -X (-2) = -3, x(-1) = 2, x(0) = 0, x(1) = 3, x(2) = 1 and x(3) = 2//This discrete-time single can a is a constant. 2. Time – Variant and Time – Invariant Systems -A system is time – invariant if its
be represented graphically as shown in Figure- characteristics and behavior are fixed over time .i.e., a time – shift in input signal causes an identical
time – shift in output signal. 𝑖𝑓 𝑥[𝑛] → 𝑦[𝑛] 𝑡ℎ𝑒𝑛, 𝑥[𝑛 − 𝑛0] → 𝑦[𝑛 − 𝑛0] ∀ 𝑛0 If the above the
relation is not satisfied, then the system is time – variant. 3. Causal and Non – causal Systems -A
system is causal or non – anticipatory or physically realizable, if the output at any time n0 depends
only on present and past inputs (n < n0), but not on future inputs. In other words, if the inputs are
equal upto some time no, the corresponding outputs must also be equal upto that time no, for a causal
Classification of Discrete – Time Sequences➔i)Energy Signals and Power Signals➔
system. 4. Stable and unstable systems -A stable system is one in which, a bounded input results in a
Functional Representation➔ In this, the amplitude of the signal is written against the values of n. response that does not diverge. Then the system is said to be BIBO stable. For a system, if the input is
The signal given in section 1.2.1 can be represented using the functional representation as follows: bounded .i.e, 𝑖𝑓 |𝑥[𝑛]| ≤ 𝑀𝑥 < ∞ ∀𝑛 And if the corresponding output is also bounded .i.e., |𝑦[𝑛]| ≤ 𝑀𝑦
< ∞ ∀𝑛 Then the system is said to be BIBO stable. 5. Memory and memoryless systems -A system
is said to possess memory, or is called a dynamic system, if its output depends on past or future values
of the input. If the output of the system depends only on the present input, the system is said to be
Tabular Representation➔ In this, the sampling instant n and the magnitude of the signal at the memoryless.
sampling instant are represented in the tabular form. The signal given in section 1.2.1 can be Frequency domain representation of discrete time signals➔ The concept of frequency is closely
represented in tabular form as follows: related to a specific type of periodic motion called harmonic oscillation, which is described by
sinusoidal functions. The CT and DT sinusoidal signals are characterized by the following properties:-
Sequence Representation➔ A finite duration sequence given in section 1.2.1 can be represented as 1. A continuous time sinusoid x(t) = cos (2πfat) is periodic for any value of fa. But for DT sinusoid
follows: X(n) = { −3,2,0,3,1,2 }↑//The arrow mark ↑ denotes the n = 0 term. When no arrow is x[n]=cos(2πfdn) to be periodic with period N (an integer), we require- cos(2𝜋𝑓𝑑𝑛) = cos[2𝜋𝑓𝑑(𝑛 +
indicated, the first term corresponds to n = 0.// So a finite duration sequence, that satisfies the 𝑁)] = cos(2𝜋𝑓𝑑𝑛 + 2𝜋𝑓𝑑𝑁) //This is possible only if 2𝜋𝑓𝑑𝑁 = 2𝜋𝑘 (𝑘 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟) //Or Fd=k/n
condition x(n) = 0 for n < 0 can be represented as: x(n) = {3, 5, 2, 1, 4, 7} //the discrete frequency fd must be a rational number (ratio of two integers). //Similarly, a discrete
Elementary discrete time sequences➔ These are the basic sequences that appear often, and play an time exponential ejωn is periodic only if 𝜔 /2𝜋 = 𝑓𝑑 = 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟. //The period is the
important role. Any arbitrary sequence can be represented in terms of these elementary sequences- denominator after 𝜔/2n is simplified such that in 𝜔 /2𝜋 = 𝑘 , k/n and N are relatively prime.// 2. A
Unit – Sample sequence➔ It is denoted by δ [ n ]. It is defined as-- CT sinusoidal signal x(t) = cos(Ωt) has a unique waveform for every value of Ω, 0 < Ω < ∞.
It is also referred as discrete time impulse. It is mathematically much ii)Periodic and aperiodic signals➔A signal x[n] is periodic with period N if and only if- 𝑥[𝑛 + 𝑁] = Increasing Ω results in a sinusoidal signal of ever – increasing frequency. //But, for a DT sinusoidal
less complicated than the continuous impulse δ (t), which is zero 𝑥[𝑛] ∀ 𝑛 The smallest N for which the above relation holds is called the fundamental period. If no signal cos (ωn), considering two frequencies separated by an integer multiple of 2π, (ω and ω + 2πm,
everywhere except at t = 0. At t = 0, it is defined in terms of its area (unit area), but not by its absolute finite value of N satisfies the above relation, the signal is said to be aperiodic or non – periodic. The m is an integer), we have- cos[(𝜔 ± 2𝜋𝑚)𝑛] = cos(𝜔𝑛 ± 2𝜋𝑚𝑛) // Since m and n are both integers-
value. It is graphically represented as-- sum of M periodic Discrete – time sequences with periods N1, N2, …, NM, is always periodic with cos(𝜔𝑛 ± 2𝜋𝑚𝑛) = cos(𝜔𝑛) So, a DT sinusoidal sequence has unique waveform only for the values of
period N where- 𝑁 = 𝐿𝐶𝑀(𝑁1, 𝑁2, … , 𝑁𝑀).iii)Even and Odd Signals ➔A real – valued discrete – ω over a range of 2π. The range −𝜋 ≤ 𝜔 ≤ 𝜋 defines the fundamental range of frequencies or principal
time signal is called an Even Signal if it is identical with its reflection about the origin .i.e., it must be range.
symmetrical about the vertical axis- 𝑥[𝑛] = 𝑥[−𝑛] ∀n //A real – valued discrete – time signal is called Frequency domain representation of discrete time systems➔ The frequency response function
an Odd Signal if it is anti symmetrical about the vertical axis. 𝑥[𝑛] = −𝑥[−𝑛] ∀𝑛// From the above completely characterizes a linear time invariant system in the frequency domain. Since, most signals
Unit step sequence➔ It is denoted by u [ n ] and defined as- relation, it can be inferred that an odd signal must be zero at time origin, n = 0. Every signal x[n] can can be expressed in Fourier domain as a weighted sum of harmonically related exponentials, the
It is graphically represented as- be expressed as the sum of its even and odd components- 𝑥[𝑛] = 𝑥𝑒[𝑛] + 𝑥𝑜[𝑛]. //Product of even and response of an LTI system to this class of signals can be easily determined. The response of any
odd sequences results in an odd sequence. • Product of two odd sequences results in an even relaxed LTI system to an arbitrary input signal x[n] is given by the convolution sum - 𝑦[𝑛] = ∑k=-∞∞
sequence. • Product of two even sequences results in an even sequence.iv)Bounded and Unbounded ℎ[𝑘]𝑥[𝑛 − 𝑘]Here, the system is characterized in the time domain by its impulse response h[n]. to
sequences ➔A discrete – time sequence x[n] is said to be bounded if each of its samples is of finite develop a frequency domain characterization of the system, we excite the system with the complex
magnitude .i.e., |𝑥[𝑛]| ≤ 𝑀𝑥 < ∞ ∀𝑛 For example, The unit step sequence u[n] is a bounded sequence, exponential 𝑥[𝑛] = 𝐴𝑒𝑗𝜔𝑛 , −∞ < 𝑛 < ∞// Where A is the amplitude and ω is any arbitrary frequency
but the sequence nu[n] is an unbounded sequence. confined to the frequency interval [ - π, π ]. By substituting this in the above convolution sum, we
Unit ramp sequence➔ It is denoted by Ur [ n ], and is defined as- Discrete – Time Systems➔ A system accepts an input such as voltage, displacement, etc. and obtain the response as- 𝑦[𝑛] = ∑ k=-∞∞ ℎ[𝑘][𝐴𝑒𝑗𝜔(𝑛−𝑘) ] // = 𝐴 [ ∑ k=-∞∞ ℎ(𝑘)𝑒 −𝑗𝜔𝑘] 𝑒 𝑗𝜔n .// Here,
It is graphically represented as- produces an output in response to this input. A system can be viewed as a process that results in the term inside the brackets is a function of frequency ω. It is the Fourier Transform of the impulse
transforming input signals into output signals.// ( Discrete - Time Input Signal, x[n]➔ Discrete - Time response h[n], and is denoted by- 𝐻(𝜔) = ∑ k=-∞∞ ℎ(𝑘)𝑒 −𝑗𝜔n //= And 𝑦[𝑛] = 𝐴𝐻(𝜔) 𝑒 𝑗𝜔n // Since
System➔Discrete - Time Output signal, y[n] ) //A discrete – time system can be represented as 𝑥[𝑛] the output differs from the input only by a constant multiplicative factor, the exponential input signal
→ 𝑦[𝑛] 𝑜𝑟, 𝑦[𝑛] = 𝑇 {𝑥[𝑛]}. is called the eigen function of the system, and the multiplicative factor is called the eigenvalue of the
system. H(ω) is a complex valued function of the frequency variable ω.

Sampling➔ Sampling is the process of recording the values of a signal at given points in time. For Nyquist Sampling Rate➔The Nyquist Sampling Rate is the lowest sampling rate that can be used Stability of an LTI system➔An LTI system is said to be stable if, for an input that is bounded, the
A/D converters, these points in time are equidistant. The number of samples taken during one second without having aliasing. The sampling rate for an analog signal must be at least two times the output of the system is also bounded for all values of n.|x[n]| for which |y[n]|<∞ //y[n] is commonly
is called the sample rate. Keep in mind that these samples are still analogue values. The mathematic bandwidth of the signal. So, for example, an audio signal with a bandwidth of 20 kHz must be known as the convolution of x[n] and the impulse response, h[n]. This can also be represented as the
description of the ideal sampling is the multiplication of the signal with a sequence of direct pulses. In sampled at least at 40 kHz to avoid aliasing. In audio CD's, the sampling rate is 44.1 kHz, which is summation expression shown below:-y[n]=x[n] x h[n]= ∑m=−∞∞ h[m]x[n-m] //this should be a finite
real A/D converters the sampling is carried out by a sample-and-hold buffer. The sample-and-hold about 10% higher than the Nyquist Sampling Rate to allow cheaper reconstruction filters to be used. value hence, the terms inside the summation should not lead to infinity which means we can come to
buffer splits the sample period in a sample time and a hold time. In case of a voltage being sampled, a The Nyquist Sampling Rate is the lowest sampling rate that can be used without having aliasing. the conclusion that-∑m=−∞∞ |h[m]|<∞ .// Thus, For the impulse response to be finite, we need to ensure
capacitor is switched to the input line during the sample time. During the hold time it is detached from Z-transform (ZT)➔ is a mathematical tool which is used to convert the difference equations in time that h[m] is absolutely integrable. This would ensure that the system will be stable. Hence, the bottom
the line and keeps its voltage. domain into the algebraic equations in z-domain.The Z-transform is a very useful tool in the analysis line is that we need an absolutely summable impulse response. This stability of a system can also be
Reconstruction➔ is the process of creating an analog voltage (or current) from samples. A digital- of a linear shift invariant (LSI) system. An LSI discrete time system is represented by difference determined using the RoC by fulfilling a couple of conditions. Conditions-The system's transfer
to-analog converter takes a series of binary numbers and recreates the voltage (or current) levels that equations. To solve these difference equations which are in time domain, they are converted first into function H(z) should include the unit circle.//Also, for a causal LTI system, all the poles should lie
corresponds to that binary number. Then this signal is filtered by a lowpass filter. This process is algebraic equations in z-domain using the Z-transform, then the algebraic equations are manipulated within the unit circle. Read on to find out more about the causality of an LTI system.
analogous to interpolating between points on a graph, but it can be shown that under certain in z-domain and the result obtained is converted back into time domain using the inverse Z- Causality of an LTI system➔If you can recall what we discussed in the previous post about
conditions the original analog signal can be reconstructed exactly from its samples. Unfortunately, the transform.The Z-transform may be of two types viz. unilateral (or one-sided) and bilateral (or two- causality, it means the output of the LTI system depends on the present and past input values of x[n].
conditions for exact reconstruction cannot be achieved in practice, and so in practice the sided).//Mathematically, if x(n)𝑥(𝑛) is a discrete-time signal or sequence, then its bilateral or two- If there is a future value, it is automatically not causal.We know that-y[n]=x[n] x h[n]
reconstruction is an approximation to the original analog signal. sided Z-transform is defined as − Z[x(n)]=X(z)=∑n=−∞∞x(n)z−n.// Where, z is a complex variable and =∑m=−∞∞h[m]x[n-,]=∑m=−∞∞ h[m]x[n-m] //when the output of y[n] is 0, for a causal function, n
Aliasing➔Aliasing is a common problem in digital media processing applications. Many readers it is given by,- z=rejw .// Where, r is the radius of a circle. Also, the unilateral or one-sided z- should be less than 0 or 0 itself.Therefore, we can conclude that for n<0 h[m] will be 0. Conditions-
have heard of "anti-aliasing" features in high-quality video cards. This page will explain what transform is defined as − Z[x(n)]=X(z)=∑n=−∞∞x(n)z−n. Causality is satisfied when the RoC falls in the region outside the outermost pole of the signal.//When
Aliasing is, and how it can be avoided.Aliasing is an effect of violating the Nyquist-Shannon Advantages of the Z-transform➔The Z-transform makes the analysis of a discrete-time system the transfer function is expressed as a ratio of the output to the input of the system, the order of the
sampling theory. During sampling the base band spectrum of the sampled signal is mirrored to every easier by converting the difference equations describing the system into simple linear algebraic numerators should be of an order lesser than the order of the denominator.
multifold of the sampling frequency. These mirrored spectra are called alias. If the signal spectrum equations.//The convolution operation in time domain is converted into multiplication in z- The Inverse Z-Transform➔The inverse Z-transform is defined as the process of finding the time
reaches farther than half the sampling frequency base band spectrum and aliases touch each other and domain.//The Z-transform exists for the signals for which the discrete-time Fourier transform (DTFT) domain signal x(n)𝑥(𝑛) from its Z-transform X(z)𝑋(𝑧). The inverse Z-transform is denoted as -
the base band spectrum gets superimposed by the first alias spectrum. The easiest way to prevent does not exist. −x(n)=Z−1[X(z)]𝑥(𝑛)=𝑍−1[𝑋(𝑧)]
aliasing is the application of a steep sloped low-pass filter with half the sampling frequency before the Limitations➔The primary limitation of the Z-transform is that using Z-transform, the frequency Since the Z-transform is defined as,-X(z)=∑n=−∞∞x(n)z−n⋅⋅⋅(1)𝑋(𝑧)=∑𝑛=−∞∞𝑥(𝑛)𝑧−𝑛
conversion. Aliasing can be avoided by keeping Fs>2Fmax. domain response cannot be obtained and cannot be plotted. Where, z is a complex variable and is given by,-z=rejω𝑧=𝑟𝑒𝑗𝜔//Where, r is the radius of a circle in z-
Anti-Aliasing➔The sampling rate for an analog signal must be at least two times as high as the Region of Convergence (ROC) of Z-Transform➔The set of points in the z-plane, for which the Z- plane. Hence, on substituting the value of z in eq. (1), we get, X(z)=X(rejω)=∑n=−∞∞[x(n)r−n]e−jωn
highest frequency in the analog signal in order to avoid aliasing. Conversely, for a fixed sampling transform of a discrete-time sequence x(n)𝑥(𝑛), that is X(z)𝑋(𝑧) converges is called the region of
rate, the highest frequency in the analog signal can be no higher than a half of the sampling rate. Any The integration given in the equation represents the integration around the circle of radius |z|=r|𝑧|=𝑟 in
convergence (ROC) of the Z-transform X(z)𝑋(𝑧).For any given discrete-time sequence, the Z- the counter clockwise direction. This is the direct method of finding inverse Z-transform. The direct
part of the signal or noise that is higher than a half of the sampling rate will cause aliasing. In order to transform may or may not converge. If there is no point in the z-plane for which the
avoid this problem, the analog signal is usually filtered by a low pass filter prior to being sampled, method is quite tedious. Hence, indirect methods are used for finding the inverse Z-transform.
function X(z)𝑋(𝑧) converges, then the sequence x(n)𝑥(𝑛) is said to be having no z-transform.
and this filter is called an anti-aliasing filter. Sometimes the reconstruction filter after a digital-to- Methods to Find the Inverse Z-Transform➔Long Division Method or Power Series Method –
Properties of ROC of Z-Transforms➔ROC of z-transform is indicated with circle in z-plane.//ROC
analog converter is also called an anti-aliasing filter. does not contain any poles.//If x(n) is a finite duration causal sequence or right sided sequence, then The long division method is simple and the advantage of this method is that it is more general and can
Converters➔As a matter of professional interest, we will use this page to briefly discuss Analog-to- the ROC is entire z-plane except at z = 0.//If x(n) is a finite duration anti-causal sequence or left sided be applied to any problem. But, the disadvantage of this method is that it does not give the solution in
Digital (A2D) and Digital-to-Analog (D2A) converters, and how they are related to the field of digital sequence, then the ROC is entire z-plane except at z = ∞. the closed form. Also, it can be used only if the region of convergence (ROC) of the given Z-
signal processing. Strictly speaking, this page is not important to the core field of DSP, so it can be Transform Analysis of Discrete-Time System➔The Z-transform plays a vital role in the design and transform X(z)𝑋(𝑧) is either of the form of |z𝑧| > 𝑎 or of the form of |z𝑧| < 𝑎.//Partial Fraction
skipped at the reader's leisure.A2D converters are the bread and butter of DSP systems. A2D analysis of discrete-time LTI (Linear Time Invariant) systems.Transfer Function of a Discrete-Time Expansion Method – In this method, the proper fraction X(z)𝑋(𝑧)⁄z𝑧 is written in terms of partial
converters change analog electrical data into digital signals through a process called sampling.D2A LTI System-The figure shows a discrete-time LTI system having an impulse response h(n)ℎ(𝑛). fractions and inverse Ztransform of each partial fraction is found by the standard Z-transform pairs and
converters attempt to create an analog output waveform from a digital signal. However, it is nearly ( Input Siquence, x[n]➔ Discrete – Time LTI System h(n)➔Output sequence, y[n] ) //Consider the then, all of them are added.//Convolution Integral Method – The convolution integral method uses
impossible to create a smooth output curve in this manner, so the output waveform is going to have a system gives an output y(n)𝑦(𝑛) for an input x(n)𝑥(𝑛). Then- y(n)=h(n)∗x(n)𝑦(𝑛)=ℎ(𝑛)∗𝑥(𝑛)//Taking the convolution property of the Z-transform and it can be used when the given Z-
wide bandwidth, and will have jagged edges. Z-transform on both the sides, we get,- Z[y(n)]=Z[h(n)∗x(n)]𝑍[𝑦(𝑛)]=𝑍[ℎ(𝑛)∗𝑥(𝑛)] //∴Y(z)=H(z)X(z) transform X(z)𝑋(𝑧) can be written as the product of two functions.
Nyquist Sampling Theorem➔If a signal is band limited and its samples are taken at sufficient rate // Therefore, the Z-transform of the impulse response h(n)ℎ(𝑛) of the system is given by, H(z) = Y(z) /
then those samples uniquely specify the signal and the signal can be reconstructed from those X(z).// Where, H(z) is called the transfer function of the discrete-time LTI system and can be defined Discrete Fourier Transforms ➔The DFT of a discrete-time signal x(n) is a finite duration discrete
samples. The condition in which this is possible is known as Nyquist sampling theorem and is derived as follows −The transfer function of a discrete time LTI system is defined as the ratio of Z-transform frequency sequence. The DFT sequence is denoted by X(k). The DFT is obtained by sampling one
below.A real signal whose spectrum is bandlimited to D Hz [X(f) = 0 for | f |>D] can be reconstructed of the output sequence to the Z-transform of the input sequence x(n)𝑥(𝑛), when the initial conditions period of the Fourier transform X(W) of the signal x(n) at a finite number of frequency points. This
from its samples taken uniformly at a rate fs > 2D samples/sec. We can say the minimum sampling are neglected. sampling is conventionally performed at N equally spaced points in the period 0 ≤w≤2w or at wk =
frequency is fs=2D Hz.// The sampled signal consists of impulses spaced every T seconds. Properties of Z-Transform➔Linearity Property➔If x(n)Z.T X(Z)//and y(n)Z.T Y(Z)//Then 2πk/N; 0 ≤ k≤ N – 1. We can say that DFT is used for transforming discrete-time sequence x(n) of
The nth impulse, located at t = nT, has a strength x(nT), the value of x(t) at t = nT,// Thus as long as linearity property states that//ax(n)+by(n)Z.T aX(Z)+bY(Z).//Time Shifting Property➔ finite length into discrete frequency sequence X(k) of finite length. The DFT is important for two
the sampling frequency fs is greater than twice the signal bandwidth D (in Hz), If x(n)⟷Z.T X(Z)//Then Time shifting property states that//x(n−m)⟷Z.T z−mX(Z). reasons. First it allows us to determine the frequency content of a signal, that is to perform spectral
X'(f) will consist of nonoverlapping repetitions of X(f). In such a case it indicates that x(t) can be Multiplication by Exponential Sequence Property-If x(n)⟷Z.T X(Z)//Then multiplication by an analysis. The second application of the DFT is to perform filtering operation in the frequency domain.
recovered from its samples x'(t) by passing the sampled signal x'(t) through an ideal low pass filter of exponential sequence property states that -an . x(n)⟷Z.T X(Z/a)Time Reversal Property➔
bandwidth D Hz. If x(n)Z.T X(Z)//Then time reversal property states that-x(−n)Z.TX(1/Z).
Discrete Fourier Series➔ The Fourier series representation o f a continuous-time periodic signal can Circular discrete convolution➔When a function gn is periodic , with period N then for functions, F FFT Algorithm-DECIMATION IN TIME (DIT) RADIX-2 FFT➔ In Decimation in time (DIT)
consist of an infinite number of frequency components, where the frequency spacing between two such that F x gn exists, the convolution is also periodic and identical to:- algorithm, the time domain sequence x(n) is decimated and smaller point DFTs are computed and
successive harmonically related frequencies is 1 / T p, and where Tp is the fundamental period. Since they are combined to get the result of N-point DFT. In general, we can say that, in DIT algorithm the
the frequency range for continuous-time signals extends infinity on both sides it is possible to have N-point DFT can be realized from two numbers of N/2-point DFTs, the N/2-point DFT can be
signals that contain an infinite number of frequency components. In contrast, the frequency range for realized from two numbers of N/4- point DFTs, and so on. In DIT radix-2 FFT, the N-point time
discrete-time signals is unique over the interval. A discrete-time signal of fundamental period N can The summation on K is called a periodic summation of the function F.// If gn is a periodic summation domain sequence is decimated into 2-point sequences and the 2-point DFT for each decimated
consist of frequency components separated by 2n / N radians. Consequently, the Fourier series of another function,g then F x gn is known as a circular convolution of F and g.// When the non-zero sequence is computed. From the results of 2-point DFTs, the 4-point DFTs, from the results of 4-point
representation o f the discrete-time periodic signal will contain at most N frequency components. This durations of both F and g are limited to the interval [0,N-1],F x gn reduces to these common forms: - DFTs, the 8-point DFTs and so on are computed until we get N-point DFT. For performing radix-2
is the basic difference between the Fourier series representations for continuous-time and discrete- FFT, the value of r should be such that, N = 2m . Here, the decimation can be performed m times,
time periodic signals. where m = log2N. In direct computation of Npoint DFT, the total number of complex additions are
Properties of DTFT➔Linearity : a1x1(n)+a2x2(n)⇔a1X1(ejω)+a2X2(ejω) N(N – 1) and the total number of complex multiplications are N 2 . In radix-2 FFT, the total number
Time shifting − x(n−k)⇔e−jωk.X(ejω) //Time Reversal − x(−n)⇔X(e−jω) of complex additions are reduced to N log2N and the total number of complex multiplications are
Frequency shifting − ejω0nx(n)⇔X(ej(ω−ω0)) reduced to (N/2) log2N.
Differentiation frequency domain − nx(n)=jd/dωX(ejω) 8-POINT DFT USIKG RADIX-2 DIT FFT➔ The computation of 8-point DFT using radix-2 FFT
Convolution − x1(n)∗x2(n)⇔X1(ejω)×X2(ejω)//Co-relation − yx1×x2(l)⇔X1(ejω)×X2(ejω) involves three stages of computation. Here N = 8 = 23 , therefore, r = 2 and m = 3. The given 8-point
Convolution➔It is a mathematical tool to combining two signals to form a third signal. Therefore, sequence is decimated into four 2-point sequences. For each 2-point sequence, the two point DFT is
in signals and systems, the convolution is very important because it relates the input signal and the computed. From the results of four 2-point DFTs, two 4-point DFTs are obtained and from the results
impulse response of the system to produce the output signal from the system. In other words, the of two 4- point DFTs, the 8-point DFT is obtained. Let the given 8-sample sequence x(n) be {x(0),
convolution is used to express the input and output relationship of an LTI system. The convolution of x(1), x(2), x(3), x(4), x(5), x(6), x(7)}. The 8-samples should be decimated into sequences of two
f and g is written f x g denoting the operator with the symbol x It is defined as the integral of the samples. Before decimation they are arranged in bit reversed order.
product of the two functions after one is reflected about the y-axis and shifted. DECIMATION IN FREQUENCY (DIF) RADIX-2 FFT➔ In decimation in frequency algorithm,
Circular convolution➔When a function gT is periodic, with period T, then for functions, f, such the frequency domain sequence X(k) is decimated. In this algorithm, the N-point time domain
that f ∗ gT exists, the convolution is also periodic and identical to: The notation F x n g for cyclic convolution denotes convolution over the cyclic group of integers sequence is converted to two numbers of N/2-point sequences. Then each N/2-point sequence is
modulo N.//Circular convolution arises most often in the context of fast convolution with a fast converted to two numbers of N/4-point sequences. This process is continued until we get N/2 numbers
Fourier transform (FFT) algorithm. of 2-point sequences. Finally, the 2-point DFT of each 2-point sequence is computed. The 2-point
FAST FOURIER TRANSFORM➔ In this section we represent several methods for computing dft DFTs of N/2 numbers of 2- point sequences will give N-samples, which is the N-point DFT of the
where t0 is an arbitrary choice. The summation is called a periodic summation of the efficiently. In view of the importance of the DFT in various digital signal processing applications such time domain sequence. Here the equations for N/2-point sequences, N/4-point sequences, etc., are
function f.//When gT is a periodic summation of another function, g, then f ∗ gT is known as as linear filtering, correlation analysis and spectrum analysis, its efficient computation is a topic that obtained by decimation of frequency domain sequences. Hence this method is called DIF.
a circular or cyclic convolution of f and g.//And if the periodic summation above is replaced by fT, the has received considerably attention by many mathematicians, engineers and scientists. Basically the
operation is called a periodic convolution of fT and gT. computation is done using the formula method.
Discrete convolution➔For complex-valued functions f, g defined on the set Z of integers,
the discrete convolution of f and g is given by: -

The convolution of two finite sequences is defined by extending the sequences to finitely supported
functions on the set of integers. When the sequences are the coefficients of two polynomials, then the
coefficients of the ordinary product of the two polynomials are the convolution of the original two
sequences. This is known as the Cauchy product of the coefficients of the sequences.

FFT algorithm exploits the two symmetry properties and so is an efficient algorithm for DFT
computation. By adopting a divide and conquer approach, a computationally efficient algorithm can
be developed. This approach depends on the decomposition of an N-point DFT into successively
smaller size DFTs.

Parseval’s Theorem of Fourier Transform➔Statement – Parseval’s theorem states that the energy DESIGN OF IIR FILTER BY BILINEAR TRSFORMATION METHOD➔ IIR filter can be
of signal x(t)𝑥(𝑡) [if x(t)𝑥(𝑡) is aperiodic] or power of signal x(t)𝑥(𝑡) [if x(t)𝑥(𝑡) is periodic] in the designed using (a) approximation of derivatives method and (b) Impulse invariant transformation
time domain is equal to the energy or power in the frequency domain. method. However the IIR filter design using these methods is appropriate only for the design of low-
Therefore, if,-x1(t)↔FTX1(ω)andx2(t)↔FTX2(ω)//Then, Parseval’s theorem of Fourier transform pass filters and band pass filters whose resonant frequencies are small. These techniques are not
states that//∫∞−∞x1(t)x∗2(t)dt=12π∫∞−∞X1(ω)X∗2(ω)dω//Where, x1(t)𝑥1(𝑡) and x2(t)𝑥2(𝑡) are suitable for high-pass or band rejects filters. The limitation is overcome in the mapping technique
complex functions. called the bilinear transformation. This transformation is a one-to-one mapping from the s-domain to
Parseval’s Identity of Fourier Transform➔The Parseval’s identity of Fourier transform states that the z-domain. That is, the bilinear transformation is a conformal mapping that transforms the
the energy content of the signal x(t)𝑥(𝑡) is given by,-E=∫∞−∞|x(t)|2dt=12π∫∞−∞|X(ω)|2dω imaginary axis of s-plane into the unit circle in the z-plane only once, thus avoiding aliasing of
The Parseval’s identity is also called energy theorem or Rayleigh’s energy theorem. frequency components. In this mapping, all points in the left half of s-plane are mapped inside the unit
The quantity [|X(ω)|2][|𝑋(𝜔)|2] is called the energy density spectrum of the signal x(t)𝑥(𝑡). circle in the z-plane, and all points in the right half of s-plane are mapped outside the unit circle in the
z-plane. So the transformation of a stable analog filter results in stable digital filter. The bilinear
transformation can be obtained by using the trapezoidal formula for the numerical integration.
DIGITAL BUTTERWORTH FILTER ➔The popular methods of designing IIR digital filter
involves the design of equivalent analog filter and then converting the analog filter to digital filter.
Digital filters➔Filters are of two types—FIR and IIR. The types of filters which make use of Hence to design a Butterworth IIR digital filter, first an analog Butterworth filter transfer function is
feedback connection to get the desired filter implementation are known as recursive filters. Their determined using the given specifications. Then the analog filter transfer function is converted to a
impulse response is of infinite duration. So they are called IIR filters. The type of filters which do not digital filter transfer function using either impulse invariant transformation or bilinear transformation.
employ any kind of feedback connection are known as non-recursive filters. Their impulse response is ANALOG BUTTERWORTH FILTER➔ The analog Butterworth filter is designed by
of finite duration. So they are called FIR filters. IIR filters are designed by considering all the infinite approximating the ideal frequency response using an error function. The error function is selected
samples of the impulse response. The impulse response is obtained by taking inverse Fourier such that the magnitude is maximally flat in the passband and monotonically decreasing in the
transform of ideal frequency response. stopband. (Strictly speaking the magnitude is maximally flat at the origin, i.e., at Ω= 0, and
monotonically decreasing with increasing Ω ).
Frequency response of the Butterworth filter➔ The frequency response of Butterworth filter
depends on the order N. The magnitude response for different values of N are shown in Figure 5.
From Figure 5, it can be observed that the approximated magnitude response approaches the ideal
response as the value of N increases. However, the phase response of the Butterworth filter becomes
more nonlinear with increasing N.
Properties of Butterworth filters➔ 1. The Butterworth filters are all pole designs (i.e. the zeros of
the filters exist at œ). //2. The filter order N completely specifies the filter.// 3. The magnitude
response approaches the ideal response as the value of N increases. //4. The magnitude is maximally
flat at the origin.// 5. The magnitude is monotonically decreasing function of Ω.
Advantages of digital filters ➔1. The values of resistors, capacitors and inductors used in analog Elliptic filters➔As sharp as a whip Has the sharpest (fastest) roll-off but has ripple in both the pass-
filters change with temperature.// 2. In digital filters, the precision of the filter depends on the length band and the stop-band.The gain for lowpass elliptic filter is given by:
(or size) of the registers used to store the filter coefficients.// 3. The digital filters are programmable.
Hence the filter coefficients can be changed any time to implement adaptive features//. 4. A single where, s is the ripple factor derived from pass-band ripple, Rn is known as nth order elliptical rational
filter can be used to process multiple signals by using the techniques of multiplexing. Disadvantages function and ξ is the selectivity factor derived from stop-band attenuation. Advantages➔Best
of digital filters➔ 1. The bandwidth of the discrete signal is limited by the sampling frequency. The selectivity among the three. Ideal for applications that want to effectively eliminate the frequencies in
bandwidth of real discrete signal is half the sampling frequency.//2. The performance of the digital the immediate neighborhood of pass-band.Disdvantages ➔Ripples in both the bands and hence, all
filter depends on the hardware (i.e., depends on the bit length of the registers in the hardware) used to frequencies experience non-identical changes in magnitude.//Non-linear phase, that leads to phase
implement the filter. Features of llR filters ➔1. The physically realizable IIR filters do not have distortion.//High complexity.
linear phase. 2. The IIR filter specifications include the desired characteristics for the magnitude Window method➔The design methods for FIR filters are based on direct approximation of the
response only. desired frequency response of the discrete-time system. For this method, the filter coefficients
Simple filters➔ In the previous lecture we considered the polynomial fit as a case example of correspond to the impulse response of the filter to be designed. To be able to process different signals,
designing a smoothing filter. The approximation to an “ideal” LPF can be improved by using a it is necessary that these signals are finite. However, this is not always the case for all signals. The
higher-degree polynomial: for example, instead of using a quadratic as in the example given in the window method is often used to create these types of finite signal sequences from infinite sequences.
previous lecture, we could have fitted a least-squares quartic to the original “noisy” data. The effect of This "cutting off" of an infinite sequence to create a finite sequence, however, affects the frequency
using a higher-degree polynomial is to give both a higher degree of tangency at ω = 0 and a sharper range. This method would be precise in principle, but since only several finite values can be used, this
cut-off in the amplitude response.All these methods though are just examples of weighted moving principle leads to deviations which primarily limit the achievable slope steepness and stopband
average filters. For example, we consider the the Hanning filter, for which:- y[k] = 1/4 (x[k]+2x[k − attenuation.The approximation of an ideal filter by cutting off the ideal impulse response is identical
1] + x[k − 2])// This filter produces an output which is a scaled average of three successive inputs, to the convergence problem in Fourier series.
with the centre point of the three weighted twice as heavily as its two adjacent neighbours.
 FIR DIGITAL FILTERS➔A filter is a frequency selective system. Digital filters are classified as Analysis filter bank➔ The D-channel analysis filter bank is shown in Figure 10.68. It consists of D
Window Functions:-The non-ideal effects, which are observed due to the finite number of filter finite duration unit impulse response (FIR) filters or infinite duration unit impulse response (IIR) sub-filters. All the subfilters are equally spaced in frequency and each have the same bandwidth. The
coefficients, can be mitigated by using a weighting window. The principle of this is that the filter filters, depending on the form of the unit impulse response of the system. In the FIR system, the spectrum of the input signal lies in the range 0 ≤ ≤ . The filter bank splits the signal into a number of
coefficients in the middle are more heavily weighted than the coefficients at the beginning and the impulse response sequence is of finite duration, i.e., it has a finite number of non-zero terms. The IIR sub-bands each having a bandwidth π /D. The filter H0(z) is a low-pass filter, H1(z) to HD–2(z) are
end. system has an infinite number of non-zero terms, i.e., its impulse response sequence is of infinite band pass and HD–1(z) is high-pass. As the spectrum of the signal is band limited to π /D, the
Filter Design Method Using the Kaiser Window➔The Kaiser window fulfills a special purpose. duration. IIR filters are usually implemented using recursive structures (feedbackpoles and zeros) and sampling rate can be reduced by a factor D. The down sampling moves all the pass band signals to a
The compromise between the width of the main lobe and the attenuation of the side lobe can be FIR filters are usually implemented using non-recursive structures (no feedback-only zeros). The base band 0 ≤ w ≤ π /D.
quantified by searching for a window maximally concentrated in the frequency range by ω = 0. Kaiser response of the FIR filter depends only on the present and past input samples, whereas for the IIR Synthesis filter bank➔ The D-channel synthesis filter bank shown in Figure 10.69 is dual of the
(1966, 1974) figured out that a virtually optimal window can be found using a modified Bessel filter, the present response is a function of the present and past values of the excitation as well as past analysis filter bank. In this case, each Vd(z) is fed to an up sampler. The up-sampling process
function of the first type and zero order.This window can be adjusted in such a way that a values of the response.Advantages of FIR filter ➔ 1. FIR filters are always stable. 2. FIR filters with produces the signal Vd(zD ). These signals are applied to filters Gd (z) and finally added to get the
compromise can be found between the amplitude of the side lobe and the width of the main lobe. As exactly linear phase can easily be designed. 3. FIR filters can be realized in both recursive and non- output signal Xˆ (z). The filters G0(z) to GD–1(z) have the same characteristics as the analysis filters
with the Chebyshev function, the filter length L can also be calculated for the Kaiser window from the recursive structures. 4. FIR filters are free of limit cycle oscillations, when implemented on a finite H0(z) to HD–1(z).
attenuation and transition data. word length digital system. 5. Excellent design methods are available for various kinds of FIR filters. Sub-band coding filter bank➔ By combining the analysis filter bank of Figure 5.25 and the
Parks–McClellan algorithm➔It is published by James McClellan and Thomas Parks in 1972, is an Disadvantages of FIR filters➔ 1. The implementation of narrow transition band FIR filters is very synthesis filter bank of Figure 5.27, we can obtain a D-channel sub-band coding filter bank shown in
iterative algorithm for finding the optimal Chebyshev finite impulse response (FIR) filter. The Parks– costly, as it requires considerably more arithmetic operations and hardware components such as Figure5.27. The analysis filter bank splits the broad band input signal x(n) into D non-overlapping
McClellan algorithm is utilized to design and implement efficient and optimal FIR filters. It uses an multipliers, adders and delay elements. 2. Memory requirement and execution time are very high. frequency band signals X0(z), X1(z), ..., XD–1(z) of equal bandwidth. These outputs are coded and
indirect method for finding the optimal filter coefficients.The goal of the algorithm is to minimize the Design OF FIR Filters using Windows➔ The procedure for designing FIR filter using windows is: transmitted. The synthesis filter bank is used to reconstruct output signal Xˆ (z) which should
error in the pass and stop bands by utilizing the Chebyshev approximation. The Parks–McClellan 1. Choose the desired frequency response of the filter Hd(ω). 2. Take inverse Fourier transform of approximate the original signal. Sub-band coding is very much used in speech signal processing.
algorithm is a variation of the Remez exchange algorithm, with the change that it is specifically Hd(ω ) to obtain the desired impulse response hd(n). 3. Choose a window sequence w(n) and multiply Subband coding and multiresolution analysis➔ A number of subband coding and multiresolution
designed for FIR filters. It has become a standard method for FIR filter design.The algorithm➔The hd(n) by w(n) to convert the infinite duration impulse response to a finite duration impulse response techniques are based on the subband decomposition and reconstruction procedure described in Section
Parks–McClellan Algorithm is implemented using the following steps:- Initialization: Choose an h(n). 4. The transfer function H(z) of the filter is obtained by taking Z-transform of h(n). 2.2. A common filter structure for subband decomposition is the filter bank in Figure 2.9, which
extremal set of frequences {ωi(0)}.//Finite Set Approximation: Calculate the best Chebyshev Parametric methods➔These methods are based on parametric models of a time series, such as AR shows a three-level decomposition scheme. The filter bank performs a number of successive lowpass
approximation on the present extremal set, giving a value δ(m) for the min-max error on the present models, moving average (MA) models, and autoregressive-moving average (ARMA) models. filtering and down-sampling operations. The signal from each stage is high-pass filtered and down-
extremal set.//Interpolation: Calculate the error function E(ω) over the entire set of frequencies Ω Therefore, parametric methods also are known as model-based methods. To estimate the PSD of a sampled. This process generates the transformed signals xH, xLH, xLLH, xLLL (for K = 3 levels),
using (2).//Look for local maxima of |E(m)(ω)| on the set Ω.//If max(ω∈Ω)|E(m)(ω)| > δ(m), then update the time series with parametric methods, you need to obtain the model parameters of the time series which have a total length equal to that of the original signal. The process shown in Figure 2.9 is called
extremal set to {ωi(m+1)} by picking new frequencies where |E(m)(ω)| has its local maxima. Make sure first.You must build an appropriate model that correctly reflects the behavior of the system that the pyramid algorithm, due to the structure of the filter bank.
that the error alternates on the ordered set of frequencies as described in (4) and (5). Return to Step 2 generates the time series; otherwise, the estimated PSD might not be reliable.The multiple signal Power Spectrum and Correlation function➔ The power spectrum of a signal gives the distribution
and iterate. classification (MUSIC) method also is a model-based spectral estimation method. of the signal power among various frequencies. The power spectrum is the Fourier transform of the
DESIGN OF LOW-PASS CHEBYSHEV FILTER ➔For designing a Chebyshev IIR digital filter, Nonparametric methods➔These methods, which include the periodogram method, Welch method, correlation function, and reveals information on the correlation structure of the signal. The strength of
first an analog filter is designed using the given specifications. Then the analog filter transfer function and Capon method, are based on the discrete Fourier transform. You do not need to obtain the the Fourier transform in signal analysis and pattern recognition is its ability to reveal spectral
is transformed to digital filter transfer function by using either impulse invariant transformation or parameters of the time series before using these methods.The primary limitation of nonparametric structures that may be used to characterise a signal. For a periodic signal, the power is concentrated in
bilinear transformation. The analog Chebyshev filter is designed by approximating the ideal methods is that the computation uses data windowing, resulting in distortion of the resulting PSDs due extremely narrow bands of frequencies, indicating the existence of structure and the predictable
frequency response using an error function. There are two types of Chebyshev approximations. In to window effects. The key benefit of nonparametric methods is the robustness—the estimated PSDs character of the signal. In the case of a pure sine wave the signal power is concentrated in one
type-1 approximation, the error function is selected such that the magnitude response is quirpele in the do not contain spurious frequency peaks. In contrast, parametric methods do not use data windowing. frequency. For a purely random signal the signal power is spread equally in the frequency domain,
passband and monotonic in the stopband. In type-2 approximation, the error function is selected such Parametric methods assume a signal fits a particular model. The estimated PSDs may contain spurious indicating the lack of structure in the signal. In general, the more correlated or predictable a signal, the
that the magnitude function is monotonic in the passband and quirpele in the stopband. The type-2 frequency peaks if the assumed model is wrong. PSDs estimated with parametric methods are less more concentrated its power spectrum, and conversely the more random or unpredictable a signal, the
magnitude response is also called inverse Chebyshev response. The type-1 design is discussed. biased and possess a lower variance than PSDs estimated with nonparametric methods if the assumed more spread its power spectrum. Therefore the power spectrum of a signal can be used to deduce the
Properties of Chebyshev filters➔ 1. The magnitude response is equiripple in the passband and model is correct. However, the magnitudes of PSDs estimated with parametric methods usually are existence of repetitive structures or correlated patterns in the signal process.
monotonic inmn the stopband. 2. The chebyshev type-1 filters are all pole designs. 3. The magnitude incorrect. STATIONARY PROCESS➔In mathematics and statistics, a stationary process is a stochastic
response approaches the ideal response as the value of N increases. Multirate digital signal processing➔ In multirate digital signal processing the sampling rate of a process whose joint probability distribution does not change when shifted in time. Consequently,
signal is changed in order to increase the efficiency of various signal processing operations. parameters such as the mean and variance, if they are present, also do not change over time and do not
Decimation, or down-sampling, reduces the sampling rate, whereas expansion, or up-sampling, follow any trends.Stationary process has the same statistical characteristics irrespective of shifts along
followed by interpolation increases the sampling rate. Some applications of multirate signal the time axis. To put it another way, an observer looking at the process from sampling time n1 would
processing are: • Up-sampling, i.e., increasing the sampling frequency, before D/A conversion in not be able to tell the difference in the statistical characteristics of the process if he moved to a
order to relax the requirements of the analog lowpass antialiasing filter. • Various systems in digital different time n2. This idea is formalised by considering the Nth order density for the process: fXn1 ,
audio signal processing often operate at different sampling rates. The connection of such systems Xn2 ,... ,XnN (xn1 , xn2 , . . , xnN )// Nth order density function for a discrete-time random process.
requires a conversion of sampling rate. • In the implementation of high-performance filtering Stationary is used as a tool in time series analysis, where the raw data is often transformed to become
operations, where a very narrow transition band is required. The requirement of narrow transition stationary; for example, economic data are often seasonal and/or dependent on a non-stationary price
bands leads to very high filter orders. level. An important type of non-stationary process that does not include a trend-like behaviour is the
cyclostationary process.

Optimal filtering ➔It is an area in which we design filters that are optimally adapted to the statistical
characteristics of a random process. As such the area can be seen as a combination of standard filter
design for deterministic signals with the random process theory of the previous section. • This
remarkable area was pioneered in the 1940’s by Norbert Wiener, who designed methods for optimal
estimation of a signal measured in noise. Specifically, consider the system in the figure below. • A
desired signal dn is observed in noise vn: xn = dn + vn • Wiener showed how to design a linear filter
which would optimally estimate dn given just the noisy observations xn and some assumptions about
the statistics of the random signal and noise processes. This class of filters, the Wiener filter, forms
the basis of many fundamental signal processing applications.
Wiener Filter➔ Wiener filtering was one of the first methods developed to reduce additive random
noise in images. It works on the assumption that additive noise is a stationary random process,
independent of pixel location; the algorithm minimizes the square error between the original and
reconstructed images. Wiener filtering is a low-pass filter, but instead of having a single cutoff Sampling Theorem➔The mathematical basis of sampling process has been laid by Nyquist sampling
frequency, it is a space-varying filter designed to use a low cutoff in low-detail regions and a high theorem. It also gives an idea about the recovery or reconstruction of the original analog signal
cutoff to retain detail in regions with edges or other high-variance features. There are several possible completely from its samples. The statement of the sampling theorem is given in two parts.//A band-
implementations for Wiener filtering. The one used in this PTC Mathcad function is the pixel-by- limited signal of finite energy which has no frequencies beyond fm(=W) Hz is completely described
pixel 2D adaptive Wiener filtering proposed by Lee in 1980 (see Two-Dimensional Signal and Image by specifying the values of the signals at the instants of time separated by 1/(2fM) seconds. //A band-
Processing, by Jae S. Lim, pages 536-40), where a space-varying filter is used, and the additive noise limited signal of finite energy which has no frequency components beyond fm Hz may be completely
is assumed to be white and zero-mean. recovered from a knowledge of its samples taken at a rate of 2f samples per second. The sampling rate
Causal & non-causal system➔ A system is called a causal system if its output depends only on the 2fM per second is called the Nyquist rate.
present and past values of the input and does not depend on future values of the input. A system is
called non-causal if the response of the system depends on the future values of that input A causal
system is also called non-anticipatory and a non-causal system is called anticipatory. Relationship between S-plane and Z-plane➔Though Laplace Transform is a very useful
Causal System- y(n) =x(n)// y(n) = 0.2 x(n) -x(n-2) // y(n-2)+y(n)= x(n) +0.5x(n-1). mathematical tool in the analysis of analog signals and systems, the analysis of any sampled signal or
Non-causal System- y(n-1)= x(n+2) //y(n)=0.4x(n-1)+x(n)-0.5x(n+1) sampled data system is extremely difficult in the frequency domain using s-plane representation. The
Phase delay➔A linear time-invariant system or device has a phase response property and a phase reason iS that the sampled signal or sampled system equations will contain infinite number of poles
delay property, where one can be calculated exactly from the other. Phase delay directly measures the and zeros. The transform technique which is conveniently used for the analysis of linear time-
device or system time delay of individual sinusoidal frequency components in the steady state.[3] If invariant discrete time system in the frequency domain is known as a very z-transform. It is powerful
the phase delay function at any given frequency—within a frequency range of interest—has the same mathematical transform is useful for tool used extensively in digital signal processing. Z solving
constant of proportionality between the phase at a selected frequency and the selected frequency linear constant coefficient difference equations just as Laplace transform is useful for solving linear
itself, the system/device will have the ideal of linear phase, which results in a constant group delay. constant coefficients differential equation. The application of the z-transform to the discrete time
Group delay➔While phase delay describes the system's response to steady state sinusoidal signals gives a rational function X(n)where z is a complex variable.
components, group delay describes the response to amplitude modulated sinusoids.The group delay is
a convenient measure of the linearity of the phase with respect to frequency in a modulation system.
For a modulation signal (passband signal), the information carried by the signal is carried exclusively
in the wave envelope. Group delay therefore operates only with the frequency components derived
from the envelope.
Odd & Even signal➔A real valued signal x(n) is called symmetric if x(-n) =x(n) //A Symmetric
signal is also called an even signal.// A Signal x(n) is called anti-symmetric if x(-n)=-x(n)// An anti-
symmetric signal is also called an odd signal. //If x(n) is odd, then x{0) = 0 x(t)= sin wt is an anti
symmetric signal while x(t)= cos wt is a symmetric signal.// A symmetric signal must be identical to
its reflection about the origin. An anti-symmetric signal is not identical to its reflection about the
origin but to its negative.
 Butterfly regarding FFT➔ FFT algorithm is shown by signal flow graph or butterfly diagram. The
FFT butterfly 1S 2 graphical method of showing multiplications and additions involving the samples.
Standard graph flow notation is used where each circle with entering arrows is a addition of the two
values at the end of the arrows multiplied by a constant. The Constant is a number which appears
beside the arrow. If there is no value then the constant is taken as unity. A butterfly is a portion of the
FFT computation that combines the results of smaller DFTs into a larger DFT or vice versa i.e.
breaking a larger DFT into sub-transforms.
Difference and similarities between DIT and DIF algorithms➔Similarities:- Both algorithms use
divide and conquer method of computation. //Both of them require N log2 N operations to compute
the DFT.// Both algorithms can be done in place. Both need to perform bit reversal at some place
during computation. Differences :-1. For DIT FFT algorithm, the input is bit-reversed but the output
is in natural order. In DIF FFT algorithm, the input is in natural order while the output is bit-reversed//
2. The DIF butterfly is slightly different from the DIT butterfly. In DIF the complex multiplication is
done after the add-subtract operation
Quantizatio➔, in mathematics and digital signal processing, is the process of mapping input values Quantization Error➔ There are basically three types of quantization errors due to finite word length
from a large set (often a continuous set) to output values in a (countable) smaller set, often with a registers in error (3) Coefficient quantization error. These errors arise due to quantization of digital
finite number of elements. Rounding and truncation are typical examples of quantization processes. filters. These are, namely, (1) Input quantization error (2) Product quantization numbers. The
Quantization is involved to some degree in nearly all digital signal processing, as the process of conversion of a continuous time input signal into digital signal produces the i-quantization error.
representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of Quantization of fitter coefficients-During digital signal processing, the filter coefficients are stored
essentially all lossy compression algorithms. in binary registers. Such registers can accommodate only a finite-number of bits. Therefore, the filter
Coetiicients have to the rounded off or truncated to accommodate them in these registers. these
rounding-off or truncation results in degradation in the system performance. Moreover, in digital
signal processing, a continuous or analog signal is first sampled and in quantized to get the digital
signal for processing. The quantization process Approximates the actual value of the signal at the
sampling instant to a discrete level which is very near to the actual value. This introduces an error
called quantization error.
Truncation and Rounding➔ Two common methods of quantization are (1) Truncation and (2)
Rounding. The effect of rounding and transaction is to introduce an error. Truncation is the process of
discarding all bits less significant than least significant bi that is retained. Let us truncate the 8-bit
binary number 0.00110011 to 4-bit binary number 0.0011. When we truncate the number, the signal
value is approximated by the highest quantization level which is not greater than the signal. Rounding
of a number of b - bits is accompanied by choosing the rounded result as the b,- bit number closest to
the original unrounded number.

Overlap-Save Method➔ The overlap save method is useful for converting a circular convolution
into a linear convolution Let the size of the input data blocks = N=L+M-1. The size of the DFTs and
IDFTS are of length N. Each data block consists of the last (M-) data points of the previous at a block.
=64. It will be followed by L new data points to form a data sequence of length N=L+M-1. An N-
point DFT is computed for each data block. The impulse response of the FIR filter is increased in
length by appending L-l zeros by zero padding. An N-point DFT of the sequence is computed once
and stored. Let H(k)}= N- point DFT of the impulse response h(n)// {Xm(k)}= DFT of the m block of
data.