The best way to prepare for Signals and Systems is to filter out the necessary information and make very short notes which can include properties of Transforms and other formulas that help in Problem Solving.
The most important thing to take care while preparing Signals and Systems is to solve a large number of problems that will teach you a number of tricks regarding GATE problems. For Problem Solving you can refer Oppenheim and Wilsky book. Otherwise, you can try solving our Question Bank in Koncept 2018 which has good problems that too fully solved.
Now coming to the topics that are covered in Signals and Systems, the following topics constitute Signals and Systems curriculum:
Continuous and Discrete Time Signals
This topic is the starting point of Signals and Systems course and is very much needed before starting other topics in the course. This is a pre-requisite for the rest of the course. You need to spend some time on this topic to gain a command over basics and these basic concepts will help you in other subjects as well. First of all, you have to study the basics such as the difference between analog, discrete time and digital signals. Then you will study about signals that most commonly appear such as Impulse Signal, Unit Step Signal, Ramp Signal, Parabolic Signal, Rectangular Function, Triangular Function, GIF, LIF, Fractional part function, Modulus Function, Signum Function, Trigonometric Functions, Exponential Functions and so on.
Then you have to study about the classification of signals into Periodic and Aperiodic Signal, Even and Odd Signal, Bounded and Unbounded Signal, Time Limited and Time Unlimited Signals, Energy and Power Signals, causal and Non-causal signals, Deterministic and Random Signals. Then you can study the graphical representation of continuous time and discrete time signals as well as the impact of operations such as Shifting, Scaling, and Reversal on the graph of a signal.
If you go through all these concepts then your basics will be very much clear and you will be able to do well in other subjects as well.
Linear Time-Invariant Systems
This probably is the most important part in Signals and Systems as all the systems are studied here only and the most important concept here is that of the impulse response and Convolution. The impulse response is used when there is a connection of more than one systems may be in cascade configuration or parallel one but we need to remember what will be the equivalent impulse response of the entire system.
Convolution is a method by which we can compute the output of a system using the impulse response and the input. People tend to skip this topic as they feel the output can be calculated in an easier way by the use of transforms rather than convolution. But you need to understand the methods of convolution such as Mathematical and Graphical Method for Continuous Time Signals, Mathematical, Graphical, Tabular and Circular Convolution for Discrete-Time Signals. If you understand these methods well then you can use them if you want to find the value of output at a particular time instant instead of the overall function as at that time these techniques are time-saving.
The next important thing the properties of Systems like Causality, Time-Invariance, Stability, Linearity and we need to study the criterion to determine each of these properties for a system because many times we may be given a system and asked which of these properties does that particular system exhibit.
Basically, there are two types of Fourier Series and Transforms as one exists for Continuous-Time Signals and other for Discrete-Time Signals and both are very similar but for EEE GATE course discrete time Fourier Series is not included and hence they only need to prepare only continuous time Fourier series.
The approach towards preparing any transform and any series is to have a three-pronged approach that is you need to remember three things which are:
- Analysis and Synthesis Equations
- Properties of Transforms
- Common Transform Pairs
The thing with the GATE exam is most of the questions are directly based on the properties of transforms and if you will calculate them completely, a lot of time will be wasted and if you apply some property, it may be done in a short time but to apply properties you need to remember some transform pairs so that you can represent a signal in terms of some known signal and apply a property, so try to prepare all the transforms in this fashion only.
The properties that are generally used include Linearity, Time Shifting, Time Reversal, Time Scaling, Differentiation in the time domain, Integration in the time domain, Convolution of signals, Multiplication of Signals, Parseval’s Theorem.
Here also the approach remains same and we just need to remember that Fourier Transform exists for Aperiodic Signals and Fourier Series for Periodic Signals and Fourier Transform approaches Fourier Series for Periodic Signals. Similar to Fourier Series in ECE GATE course there is Discrete-Time Fourier Transform whereas in EE GATE course only Continuous-Time Fourier Transform is included.
Also, there is one very important duality that a signal discrete in one domain is periodic in other and vice versa so a periodic signal in time domain is discrete in the frequency domain which means Fourier series exists as Fourier series exists only for periodic signals and that too provided Drichlet Conditions are satisfied.
The most confusing thing that students face in this chapter is the Fourier Transform of Rectangular and Triangular Functions which involve sinc functions. But if you remember the Transform Pair for these functions once then you can easily compute the Transform. Also, there is an important property in Fourier transform which is the Duality Property and to apply this you should remember some of the common Transform pairs.
There are certain additional concepts that ECE Aspirants need to prepare in Continuous Time Fourier Transform which are Distortion-less Transmission in LTI system, Group and Phase Delay, Cross-Correlation and Auto-Correlation. The same thing needs to be repeated in Discrete Time Fourier Transform (DTFT) but remember that since the signal is discrete in the time domain, it will be periodic in Frequency Domain. You also need to study about correlation in DTFT as well.
Laplace Transforms only exist for Continuous Time signals and is the most important transform as it is used in Engineering Mathematics and Control Systems as well and the only additional thing that comes here is the concept of Region of Convergence (ROC) as the same transform may have different inverses based on different ROCs. So please remember no Laplace Transform is complete without ROC and rest of the approach remains same as we need to follow three-pronged approach.
One more thing that is introduced here is the concept of Initial Value and Final Value Theorem and one must always remember that final value only exists subject to the condition that the system is stable and sometimes students may be fooled by this, so before applying final value theorem, please verify the stability.
Z-Transform is the analogue of Laplace Transform but for the discrete time signals and hence the properties are very similar but there are some striking differences also like in case of Final Value Theorem so such differences must be clearly remembered but rest of the things and even concept of ROC remains the same.
One thing to be noticed in Laplace and Z Transform is the concept of Stability and Causality as sometimes that may come in handy if pole-zero plot is given and system properties need to be identified.
In case of z-transform, the process of computation of inverse transform is important so you need to study about three methods which are Partial Fraction, Long Division Method, Cauchy’s Residue Method.
The only important concept in Sampling is for the Nyquist Rate and Nyquist Frequency and you must practice drawing one or two waveforms where the sampling frequency is less than Nyquist Frequency and that will result in aliasing as sometimes waveforms may also be asked. You may come across many weird kinds of problems, please ignore them as they are not important for GATE but please also read about Band-Pass Sampling Theorem as that may also be asked. But to be good at problems based on Sampling theory you need to have a good command of Fourier transform first.
These topics are a further extension of Fourier Transform but are not included in EE curriculum. In DFT you again need to study the computation of DFT using Analysis Equation. There are certain shorter techniques as well such as Matrix Method of computation of DFT so you need to study about that also. Again you need to prepare the properties well. You can expect at least one problem in GATE from this topic.
In case of FFT (Fast Fourier Transform) you need to first understand the complexity of operations involved in DFT such as how many complex additions are required, the number of complex multiplications required and so on. then you need to study about the Decimation in time and Decimation in Frequency Concept for radix-2 FFT algorithm. Here you need to understand the concept of Butterfly and how it helps in computation.