Continuous-Time System Properties Linear Systems and Signals Lecture 2 Spring 2008

2 - 2 Linearity A system is linear if it is both –Homogeneous: If we scale the input, then the output is scaled by the same amount: –Additive: If we add two input signals, then the output will be the sum of their respective outputs Response of a linear system to zero input?

2 - 3 Examples Identity system. Linear? Ideal delay by T seconds. Linear? Scale by a constant (a.k.a. gain block). Linear? x(t)x(t) y(t)y(t) x(t)x(t)y(t)y(t)x(t)x(t)y(t)y(t) Two different but equivalent graphical syntaxes x(t)x(t) y(t)y(t)

2 - 4  Each T represents a delay of T time units Examples Tapped delay line Linear? There are N-1 delays … …

2 - 5 Examples Transcendental system Answer: Nonlinear (in fact, fails both tests) Squarer Answer: Nonlinear (in fact, fails both tests) Differentiation is linear –Homogeneity test: –Additivity test: x(t)x(t) y(t)y(t)

2 - 6 Examples Integration –Homogeneity test –Additivity test Answer: Linear Human hearing –Responds to intensity on a logarithmic scale Answer: Nonlinear (in fact, fails both tests) x(t)x(t) y(t)y(t)

2 - 7 Examples Human vision –Similar to hearing in that we respond to the intensity of light in visual scenes on a logarithmic scale. Answer: Nonlinear (in fact, fails both tests) Modulation by time Answer: Linear

2 - 8 Examples Amplitude Modulation (AM) but not AM radio y(t) = A x(t) cos(2  f c t) f c is the carrier frequency (frequency of radio station) A is a constant Answer: Linear A x(t)x(t) cos(2  f c t) y(t)y(t)

2 - 9 Examples Frequency Modulation (FM) –FM radio: –f c is the carrier frequency (frequency of radio station) –A and k f are constants Answer: Nonlinear (fails both tests) + kfkf x(t)x(t) A 2fct2fct Linear NonlinearLinear y(t)y(t)

Time-Invariance A system is time-invariant if –When the input is shifted in time, then its output is shifted by the same amount –This must hold for all possible shifts. If a shift in input x(t) by t 0 causes a shift in output y(t) by t 0 for all real-valued t 0, then system is time-invariant: x(t)x(t) y(t)y(t) x(t – t 0 ) y shifted (t) Does y shifted (t) = y(t – t 0 ) ?

Examples Identity system –Step 1: compute y shifted (t) = x(t – t 0 ) –Step 2: does y shifted (t) = y(t – t 0 ) ? YES. Answer: Time-invariant Tapped Delay Line Answer: Time-invariant

Examples Transcendental system Answer: Time-invariant Squarer Answer: Time-invariant Differentiator Answer: Time-invariant

Examples Integration Answer: Time-invariant Human hearing Answer: Time-invariant Human vision Answer: Spatially-varying

Examples Amplitude modulation (not AM radio) FM radio + kfkf x(t)x(t) A 2fct2fct Time- invariant Time- varying Time- invariant y(t)y(t) A cos(2  f c t) Time- invariant Time- varying x(t)x(t)y(t)y(t)

Memoryless A mathematical description of a system may be memoryless, but an implementation of a system may use memory.

Example #1 Differentiation –A derivative computes an instantaneous rate of change. Ideally, it does not seem to depend on what x(t) does at other instances of t than the instant being evaluated. –However, recall definition of a derivative: –What happens at a point of discontinuity? We could average left and right limits. –As a system, differentiation is not memoryless. Any implementation of a differentiator would need memory. t x(t)x(t)

Example #2 Analog-to-digital conversion –Lecture 1 mentioned that A/D conversion would perform the following operations: –Lowpass filter requires memory –Quantizer is ideally memoryless, but an implementation may not be quantizer lowpass filter Sampler 1/T

Causality System is causal if output depends on current and previous inputs and previous outputs When a system works in a time domain, causality is generally required For images, causality is not an issue when the entire image is available because we could process pixels from upper left-hand corner to lower right-hand corner, or vice-versa

Summary If several causes are acting on a linear system, then the total effect is the sum of the responses from each cause In time-invariant systems, system parameters do not change with time For memoryless systems, the system response at any instant t depends only on the present value of the input (value at t)

Summary If a system response at t depends on future input values (beyond t), then the system is noncausal A signal defined for a continuum of values of the independent variable (such as time) is a continuous-time signal A signal whose amplitude can take on any value in a continuous range is an analog signal