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Signals and Systems

Signals and Systems

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Complex Exponential Fourier Series (Solved MCQ 2)

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Complex Exponential Fourier Series (Solved MCQ 2)

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Classroom Contents

Signals and Systems

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  1. 1 Introduction to Signals and Systems
  2. 2 Continuous and Discrete Time Signals
  3. 3 Addition of Continuous-Time Signals
  4. 4 Multiplication of Continuous-Time Signals
  5. 5 Time Scaling of Continuous-Time Signals
  6. 6 Amplitude Scaling of Continuous-Time Signals
  7. 7 Time Shifting of Continuous-Time Signals
  8. 8 Amplitude Shifting of Continuous-Time Signals
  9. 9 Reversal of Continuous-Time Signals
  10. 10 Multiple Transformations of Continuous-Time Signals
  11. 11 Multiple Transformations of CTS (Important Point & Shortcut)
  12. 12 Multiple Transformations of CTS (Solved Problem 1)
  13. 13 Multiple Transformations of CTS (Solved Problem 2)
  14. 14 Even and Odd Signals
  15. 15 Even and Odd Components of a Signal
  16. 16 Properties of Even and Odd Signals
  17. 17 Even and Odd Signals (Solved Problem 1)
  18. 18 Even and Odd Signals (Solved Problem 2)
  19. 19 Even and Odd Signals (Solved Problem 3)
  20. 20 Even and Odd Signals (Solved Problem 4)
  21. 21 Periodic and Non-Periodic Signals
  22. 22 Periodic and Non-Periodic Signals (Important Point)
  23. 23 Calculation of Fundamental Period
  24. 24 Periodic and Non-Periodic Signals (Solved Problems)
  25. 25 Effect of Time-Shifting & Time-Scaling on Fundamental Time Period
  26. 26 Effect of Time-Reversal and Phase Shift on Fundamental Time Period
  27. 27 Effect of Amplitude-Shifting on Fundamental Time Period
  28. 28 Effect of Amplitude-Scaling on Fundamental Time Period
  29. 29 Effect of Amplitude Reversal on Fundamental Time Period
  30. 30 Conjugate Symmetric Signals
  31. 31 Conjugate Antisymmetric Signals
  32. 32 Half-Wave Symmetric Signals
  33. 33 Half Wave Symmetric Signals (Solved Problems)
  34. 34 Area of Continuous-Time Signals
  35. 35 Average Value of Continuous-Time Signals
  36. 36 Average Value of Continuous-Time Signals (Solved Problems) | Part 1
  37. 37 Average Value of Continuous-Time Signals (Solved Problems) | Part 2
  38. 38 Average Value of Continuous Time Signals (Solved Problems) | Part 3
  39. 39 Energy and Power of Continuous Time Signals
  40. 40 Energy Signals
  41. 41 Energy of CT Signals (Solved Problems) | Part 1
  42. 42 Energy of CT Signals (Solved Problems) | Part 2
  43. 43 Energy of CT Signals (Solved Problems) | Part 3
  44. 44 Power Signals
  45. 45 Power of CT Signals (Solved Problems) | Part 1
  46. 46 Power of CT Signals (Solved Problems) | Part 2
  47. 47 Power of CT Signals (Solved Problems) | Part 3
  48. 48 Power, RMS & Average Values of Standard Signals
  49. 49 Neither Energy Nor Power (NENP) Signals
  50. 50 Shortcut for Energy, Power & NENP Signals
  51. 51 Energy, Power & NENP Signals (Solved Problem 1)
  52. 52 Energy, Power & NENP Signals (Solved Problem 2)
  53. 53 Energy, Power & NENP Signals (Solved Problem 3)
  54. 54 Energy and Power of Discrete-Time Signals
  55. 55 Orthogonal Signals
  56. 56 Orthogonal Signals (Solved Problems 1)
  57. 57 Orthogonal Signals (Solved Problems 2)
  58. 58 Orthogonal Signals (Solved Problems 3)
  59. 59 Unit Impulse Signal
  60. 60 Properties of Impulse Signal (Part 1)
  61. 61 Properties of Impulse Signal (Part 2)
  62. 62 Impulse Signal (Solved Problems) | Part 1
  63. 63 Impulse Signal (Solved Problems) | Part 2
  64. 64 Unit Step Signal
  65. 65 Unit Ramp Signal
  66. 66 Unit Parabolic Signal
  67. 67 Relation between Unit Impulse and Unit Step Signals
  68. 68 Signum Function
  69. 69 Unit Rectangular Function
  70. 70 Unit Triangular Function
  71. 71 Sinc Function
  72. 72 Sampling Property of Unit Impulse Signal
  73. 73 Exponential Signals (Real and Complex)
  74. 74 Mathematical Representation of Signal Waveforms (Shortcut)
  75. 75 Mathematical Representation of Signal Waveforms (Solved Problems)
  76. 76 Differentiation of Continuous-Time Signals
  77. 77 Differentiation of Continuous Time Signals (Solved Problems)
  78. 78 Integration of Continuous-Time Signals
  79. 79 Integration of Continuous-Time Signals (Solved Problem)
  80. 80 Graphical Integration - Examples (Part 1)
  81. 81 Graphical Integration - Examples (Part 2)
  82. 82 Basic System Properties
  83. 83 Static and Dynamic Systems
  84. 84 Static and Dynamic Systems (Solved Problems) | Part 1
  85. 85 Static and Dynamic Systems (Solved Problems) | Part 2
  86. 86 Causal and Non-Causal Systems
  87. 87 Causal and Non-Causal Systems (Solved Problems) | Part 1
  88. 88 Causal and Non-Causal Systems (Solved Problems) | Part 2
  89. 89 Time-Invariant and Time-Variant Systems
  90. 90 Time-Invariant and Time-Variant Systems (Solved Problems) | Part 1
  91. 91 Time-Invariant and Time-Variant Systems (Solved Problems) | Part 2
  92. 92 Time-Invariant and Time-Variant Systems (Split System)
  93. 93 Linear and Non-Linear Systems
  94. 94 Linear and Non-Linear Systems (Solved Problems) | Part 1
  95. 95 Linear and Non-Linear Systems (Solved Problems) | Part 2
  96. 96 Linear and Non-Linear Systems (Solved Problems) | Part 3
  97. 97 Linear and Non-Linear Systems (Solved Problems) | Part 4
  98. 98 Linear and Non-Linear Systems (Integral & Differential Operators)
  99. 99 Linear and Non-Linear Systems (Even & Odd Operators)
  100. 100 Linear and Non-Linear Systems (Real & Imaginary Operators)
  101. 101 Linear and Non-Linear Systems (Various Nonlinear Operators)
  102. 102 Linear and Non-Linear Systems (All Properties)
  103. 103 Checking System Linearity using Transfer Characteristics
  104. 104 Invertible and Non-Invertible Systems
  105. 105 Invertible and Non-Invertible Systems (Solved Problems)
  106. 106 Stable and Unstable Systems
  107. 107 Stable and Unstable Systems (Solved Problems) | Part 1
  108. 108 Stable and Unstable Systems (Solved Problems) | Part 2
  109. 109 Prerequisites for LTI Systems (Revision of Linearity & Time Variance)
  110. 110 Prerequisites for LTI Systems (Laplace Transform)
  111. 111 Linear Time-Invariant (LTI) Systems
  112. 112 Transfer Function
  113. 113 Transfer Function Calculation
  114. 114 Impulse Response and Convolution
  115. 115 Introduction to Convolution Operation
  116. 116 Properties of Convolution (Part 1)
  117. 117 Properties of Convolution (Part 2)
  118. 118 Properties of Convolution (Part 3)
  119. 119 Properties of Convolution (Part 4)
  120. 120 Convolution Operation (Shortcut)
  121. 121 Convolution (Solved Problem 1)
  122. 122 Convolution (Solved Problem 2)
  123. 123 Convolution (Solved Problem 3)
  124. 124 Convolution (Solved Problem 4)
  125. 125 Convolution (Solved Problem 5)
  126. 126 Convolution (Solved Problem 6)
  127. 127 Static LTI System
  128. 128 Static LTI System (Solved Problems)
  129. 129 Causal LTI System
  130. 130 Causal LTI System (Solved Problems)
  131. 131 Stable LTI System
  132. 132 Stable LTI System (Solved Problems) | Part 1
  133. 133 Stable LTI System (Solved Problems) | Part 2
  134. 134 Harmonics and LTI System
  135. 135 Magnitude Distortion in LTI Systems
  136. 136 Phase Distortion in LTI Systems
  137. 137 Distortionless LTI System
  138. 138 Group Delay and Phase Delay
  139. 139 Group Delay & Phase Delay (Solved Problems)
  140. 140 Standard Differential Equation for LTI Systems
  141. 141 Introduction to Fourier Series
  142. 142 Conditions for Existence of Fourier Series (Dirichlet Conditions)
  143. 143 Trigonometric Fourier Series
  144. 144 Trigonometric Fourier Series (Example 1)
  145. 145 Trigonometric Fourier Series (Example 2)
  146. 146 Trigonometric Fourier Series (Example 3)
  147. 147 Trigonometric Fourier Series (Example 4)
  148. 148 Trigonometric Fourier Series (Example 5)
  149. 149 Complex Exponential Fourier Series
  150. 150 Complex Exponential Fourier Series (Example 1)
  151. 151 Complex Exponential Fourier Series (Example 2)
  152. 152 Complex Exponential Fourier Series (Example 3)
  153. 153 Complex Exponential Fourier Series (Example 4)
  154. 154 Complex Exponential Fourier Series (Example 5)
  155. 155 Relation between Fourier Coefficients
  156. 156 Relation between Fourier Coefficients (Solved Problem)
  157. 157 x(t)−Cn Pairs (Shortcut for Objectives)
  158. 158 Complex Exponential Fourier Series (Solved MCQ 1)
  159. 159 Complex Exponential Fourier Series (Solved MCQ 2)
  160. 160 Complex Exponential Fourier Series (Solved MCQ 3)
  161. 161 Symmetricities in Fourier Series (Part 1)
  162. 162 Symmetricities in Fourier Series (Part 2)
  163. 163 Hidden Symmetry in Fourier Series (Examples)
  164. 164 Properties of Fourier Series (Part 1)
  165. 165 Properties of Fourier Series (Part 2)
  166. 166 Properties of Fourier Series (Part 3)
  167. 167 Properties of Fourier Series (Part 4)
  168. 168 Properties of Fourier Series (Part 5)
  169. 169 Properties of Fourier Series (Part 6)
  170. 170 Parseval's Power Theorem
  171. 171 Parseval's Power Theorem (Solved Example)
  172. 172 Properties of Fourier Series (Solved Problems)
  173. 173 Fourier Series for LTI Systems
  174. 174 Introduction to Fourier Transform
  175. 175 Conditions for Existence of Fourier Transform (Dirichlet Conditions)
  176. 176 Properties of Fourier Transform (Part 1)
  177. 177 Properties of Fourier Transform (Part 2)
  178. 178 Properties of Fourier Transform (Part 3)
  179. 179 Properties of Fourier Transform (Part 4)
  180. 180 Properties of Fourier Transform (Part 5)
  181. 181 Properties of Fourier Transform (Part 6)
  182. 182 Properties of Fourier Transform (Part 7)
  183. 183 Properties of Fourier Transform (Part 8)
  184. 184 Parseval's Energy Theorem
  185. 185 Duality Property of Fourier Transform
  186. 186 Fourier Transform of Basic Signals (DC Value)
  187. 187 Fourier Transform of Basic Signals (Impulse Signal)
  188. 188 Fourier Transform of Basic Signals (Exponential Signals)
  189. 189 Fourier Transform of Basic Signals (Signum Function)
  190. 190 Fourier Transform of Basic Signals (Unit Step Signal)
  191. 191 Fourier Transform of sgn(t) & u(t) [Important Shortcut]
  192. 192 Fourier Transform of Basic Signals (Complex Exponential Signal)
  193. 193 Fourier Transform of Basic Signals (Cosω₀t)
  194. 194 Fourier Transform of Basic Signals (Sinω₀t)
  195. 195 Fourier Transform of Basic Signals (Rectangular Function)
  196. 196 Fourier Transform of Basic Signals (Triangular Function)
  197. 197 Fourier Transform of Basic Signals (Sampling Function)
  198. 198 Fourier Transform (Solved Problem 1)
  199. 199 Fourier Transform (Solved Problem 2)
  200. 200 Fourier Transform (Solved Problem 3)
  201. 201 Fourier Transform (Solved Problem 4)
  202. 202 Fourier Transform (Solved Problem 5)
  203. 203 Fourier Transform (Solved Problem 6)
  204. 204 Fourier Transform (Solved Problem 7)
  205. 205 Fourier Transform (Solved Problem 8)
  206. 206 Fourier Transform (Solved Problem 9)
  207. 207 Fourier Transform (Solved Problem 10)
  208. 208 Calculation of Cn using Fourier Transform
  209. 209 Fourier Transform for Periodic Signals
  210. 210 Fourier Transform (Solved Problem 11)
  211. 211 x(t)−X(ω) Pairs (Shortcut for Objectives)
  212. 212 Fourier Transform (Solved Problem 12)
  213. 213 Fourier Transform (Solved Problem 13)
  214. 214 Fourier Transform (Solved Problem 14)
  215. 215 Fourier Transform (Solved Problem 15)
  216. 216 Fourier Transform (Solved Problem 16)
  217. 217 Fourier Transform (Solved Problem 17)
  218. 218 Fourier Transform (Solved Problem 18)
  219. 219 Fourier Transform (Solved Problem 19)
  220. 220 Fourier Transform (Solved Problem 20)
  221. 221 Fourier Transform (Solved Problem 21)
  222. 222 Fourier Transform (Solved Problems 22 & 23)
  223. 223 Introduction to Laplace Transform
  224. 224 Relation between Laplace Transform & Fourier Transform
  225. 225 Condition for Existence of Laplace Transform
  226. 226 ROC and its Properties
  227. 227 ROC and Stability
  228. 228 Shortcut for Region of Convergence (ROC)
  229. 229 Properties of Laplace Transform (Part 1)
  230. 230 Properties of Laplace Transform (Part 2)
  231. 231 Properties of Laplace Transform (Part 3)
  232. 232 Properties of Laplace Transform (Part 4)
  233. 233 Properties of Laplace Transform (Part 5)
  234. 234 Properties of Laplace Transform (Part 6)
  235. 235 Initial Value Theorem
  236. 236 Final Value Theorem
  237. 237 Laplace Transform of Basic Signals (Unit Impulse Signal)
  238. 238 Laplace Transform of Basic Signals (Unit Step Signal)
  239. 239 Laplace Transform of Basic Signals (DC Value & Signum Function)
  240. 240 Laplace Transform of Basic Signals (Exponential Signals)
  241. 241 Laplace Transform of Basic Signals (Unit Ramp Signal)
  242. 242 Laplace Transform of Basic Signals [Cosω₀t.u(t)]
  243. 243 Laplace Transform of Basic Signals [Sinω₀t.u(t)]
  244. 244 Laplace Transform (Solved Problem 1)
  245. 245 Laplace Transform (Solved Problem 2)
  246. 246 Laplace Transform (Solved Problem 3)
  247. 247 Laplace Transform (Solved Problem 4)
  248. 248 Laplace Transform (Solved Problem 5)
  249. 249 Laplace Transform (Solved Problem 6)
  250. 250 Laplace Transform (Solved Problem 7)
  251. 251 Laplace Transform (Solved Problems 8 & 9)
  252. 252 Laplace Transform (Solved Problem 10)
  253. 253 Laplace Transform (Solved Problem 11)
  254. 254 Inverse Laplace Transform
  255. 255 Inverse Laplace Transform (Solved Problem)
  256. 256 ROC and Stability of Causal LTI Systems
  257. 257 Laplace Transform to Solve Differential Equations
  258. 258 Laplace Transform (Solved Problem 12)
  259. 259 Laplace Transform (Solved Problem 13)
  260. 260 Laplace Transform (Solved Problem 14)
  261. 261 Sampling Theorem
  262. 262 Nyquist Rate & Nyquist Interval
  263. 263 Properties of Nyquist Rate
  264. 264 Nyquist Rate (Solved Problem 1)
  265. 265 Nyquist Rate (Solved Problem 2)
  266. 266 Nyquist Rate (Solved Problem 3)
  267. 267 Reconstruction of Signals
  268. 268 Reconstruction of a Signal Using Practical LPF
  269. 269 Frequency Components Present in S(f)
  270. 270 Reconstruction of Signals (Solved Problem 2)
  271. 271 Reconstruction of Signals (Solved Problem 3)
  272. 272 Sampling Theorem for Band Pass Signals
  273. 273 Operations on Discrete Time Signals (Time Shifting)
  274. 274 Operations on DTS (Time Compression, Time Expansion & Time Reversal)
  275. 275 Operations on DTS (Amplitude Scaling & Amplitude Reversal)
  276. 276 Addition of Discrete Time Signals
  277. 277 Multiplication of Discrete Time Signals
  278. 278 Multiple Operations on Discrete Time Signals (Part 1)
  279. 279 Multiple Operations on Discrete Time Signals (Part 2)
  280. 280 Even and Odd Discrete Time Signals
  281. 281 Even and Odd Parts of Discrete Time Signal
  282. 282 Conjugate Symmetric and Conjugate Antisymmetric DTS
  283. 283 Periodic Discrete Time Signals
  284. 284 Periodic Discrete Time Signals (Solved Problems)
  285. 285 Discrete Time Energy Signals
  286. 286 Discrete Time Energy Signals (Solved Problems)
  287. 287 Discrete Time Power Signals
  288. 288 Discrete Time Power Signals (Solved Problems)
  289. 289 Unit Impulse Sequence
  290. 290 Unit Step & Unit Ramp Sequences
  291. 291 Discrete Time Signum Function
  292. 292 Discrete Time Convolution
  293. 293 Discrete Time Convolution (Tabular Method)
  294. 294 Static & Dynamic Discrete Time Systems
  295. 295 Causal and Non-Causal Discrete Time Systems
  296. 296 Time−Invariant & Time−Variant Discrete Time Systems
  297. 297 Linear and Non−Linear Discrete Time Systems
  298. 298 Stable & Unstable Discrete Time Systems
  299. 299 Discrete Time Signals & Systems (Solved Problem 1)
  300. 300 Discrete Time Signals & Systems (Solved Problem 2)
  301. 301 Discrete Time Signals & Systems (Solved Problem 3)
  302. 302 Discrete Time Signals & Systems (Solved Problem 4)
  303. 303 Discrete Time Signals & Systems (Solved Problem 5)
  304. 304 Discrete Time Signals & Systems (Solved Problem 6)
  305. 305 Introduction to Z-Transform
  306. 306 Condition for Existence of Z-Transform
  307. 307 Properties of Z-Transform (Part 1)
  308. 308 Properties of Z-Transform (Part 2)
  309. 309 Properties of Z-Transform (Part 3)
  310. 310 Properties of Z-Transform (Part 4)
  311. 311 Properties of Z-Transform (Part 5)
  312. 312 Properties of Z-Transform (Initial Value Theorem)
  313. 313 Properties of Z-Transform (Final Value Theorem)

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