EECE 340 Homework, Spring 2008

American University of Beirut Department of Electrical and Computer Engineering EECE 440 Signals and Systems Homework 1: Due Tuesday July 4, 2006 Part I 1. For each of the following signals, determine the energy, the mean value, the average power, and the rms value of the following signals. a. ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ≥ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ π = Otherwise 0 0 t T cos 2 t )t( x 0 1 b. ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ≤ − − − < < = Otherwise 0 1 5 t 5 t 1 1 t 0 1 x 2 )t( c. ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ≤ ≤ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ π = Otherwise 0 T t 0 T 2 t cos )t( x 0 0 2 3 d. 2 t 4 3e 7 )t( x − + = 2. Determine whether or not each of the following continuous-time signals is periodic. If the signal is periodic, determine its fundamental period. a. ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + π = 3 cos 4t x )t( b. )1 t (je )t( x π − = c. 2 3 cos 2t x )t( ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − π = 3. In Chapter 1, we introduced a number of general properties of systems. In particular, a system may or may not be: a. Memoryless b. Time invariant c. Linear d. Causal e. Stable Determine which of these properties hold and which do not hold for each of the following continuous-time systems. Justify your answers. In each example, y(t) denotes the system output and x(t) is the system input. a. )t x(2 2) x t( y )t( − + − = b. )t3 x )t( (cos y )t( = c. ∫ −∞ τ τ = t x( )d y )t( d. ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≥ − − < = 0 t 2) x t( x )t( 0 t 0 y )t( e. 3) /t( x y )t( = Part II 1. An exponentially damped sinusoidal signal is defined by: at /3 e) 1000t 20sin(2 )t( x − − π π× = Where the exponential parameter a is variable, taking on the set of values a=500, 750, and 1000. Using MATLAB, investigate the effect of varying a on the signal x(t) for 2 t 2 ≤ ≤ − milliseconds.