#18 Signals & Systems #18 | Systems & Their Properties - LTI Systems

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. System and domain

   Corrected solution notes. The audio is the original recording and may contain errors or incomplete phrases. Where they differ, use the corrected written solution.

   A system maps input signals to output signals on a specified domain.

   Memory, causality, linearity, time invariance, stability and invertibility are separate properties, not a complete numerical model by themselves.

   Narration transcript

   A system takes one or more input signals, processes them, and produces one or more output signals. For continuous-time systems, we write x of t as the input and y of t as the output, connected by the operator H. The notation y of t equals H applied to x of t means the system transforms the entire input signal into the output. For discrete-time systems, the same idea holds: x of n maps to y of n through the operator S. Systems can be as simple as a multiplier or as complex as an audio equalizer — what matters is that we can fully characterize their behavior through these six properties.

2. 2. Memory and causality

   A memoryless output uses only the current input. A causal output uses only present and past inputs.

   A delay has memory and is causal. A future advance is noncausal as an operator on arbitrary live inputs, though stored data can be processed offline.

   Narration transcript

   A system is memoryless if the output at any time depends only on the input at that same time. For example, y of t equals two x of t is memoryless — to compute the output now, we only need the input now. In contrast, y of n equals x of n minus one has memory because the output depends on a past input sample. Accumulators and integrators always have memory. Causality is related: a causal system's output depends only on present and past inputs, never future ones. Every memoryless system is automatically causal. A non-causal system, like y of t equals x of t plus one, would need to know future inputs — this is impossible to implement in real time. All real-time physical systems are causal.

3. 3. Superposition

   Linearity requires H(ax1+bx2)=aH(x1)+bH(x2) for all allowed signals and scalars.

   Squaring the input fails this test in general. Fixed nonzero initial-state contributions must also be considered.

   Narration transcript

   Linearity is perhaps the most powerful system property. A system H is linear if it satisfies two conditions: homogeneity and additivity — together called superposition. Homogeneity means scaling the input scales the output by the same factor: H applied to a times x of t equals a times H applied to x of t. Additivity means the response to a sum of inputs equals the sum of the individual responses: H applied to x one plus x two equals H of x one plus H of x two. Combined, superposition states: H applied to a x one plus b x two equals a H of x one plus b H of x two. A non-linear example: y of t equals x squared of t. Check: H of a x equals a squared x squared, but a times H of x equals a times x squared. These are not equal, so the system is nonlinear.

4. 4. Time-shift test

   Compare applying the system after shifting the input with shifting the original output.

   For y[n]=n x[n], the coefficients n and n−n0 differ in the two paths, so the system is time-varying.

   Narration transcript

   A system is time-invariant if shifting the input in time produces the same shift in the output. Formally: if y of t is the response to x of t, then the response to x of t minus t zero must be y of t minus t zero. We use a two-path test to verify this. Path one: shift the input first, then apply the system. Path two: apply the system first, then shift the output. If both paths give the same result for every shift t zero, the system is time-invariant. Example: y of n equals n times x of n is time-varying, because the coefficient n changes with time. Shifting the input gives n times x of n minus n zero, but shifting the output gives n minus n zero times x of n minus n zero. These are different.

5. 5. Stability and invertibility

   BIBO stability requires every bounded input to produce a bounded output. Invertibility requires distinct allowed inputs to have distinct outputs.

   Multiplication by two is invertible; squaring arbitrary signed real signals is not. Multiplication by exp(t) is unstable on an unbounded future time axis.

   Narration transcript

   A system is BIBO stable — bounded-input bounded-output stable — if every bounded input produces a bounded output. If the input satisfies the absolute value of x of t less than or equal to M x for all t, then the output must satisfy the absolute value of y of t less than or equal to M y for some finite M y. For example, y of t equals e to the t times x of t is unstable: even a bounded input like x of t equals one produces an output that grows without bound. Invertibility means a system has a unique inverse: different inputs always produce different outputs. An invertible system is one-to-one. Example: y of t equals two x of t is invertible, with inverse system x of t equals one half y of t. Non-invertible example: y of t equals x squared of t — both x of t and negative x of t produce the same output, so we cannot recover the input.

6. 6. Correct the two-sample example

   y[n]=x[n]+x[n−1] has memory and is causal, linear, time-invariant and BIBO stable, with output bound 2M for input bound M.

   It is a sum; a normalized average additionally divides by two.

   On bilateral sequences it is not injective: zero input and x[n]=(−1)ⁿ both produce zero output.

   Forward recursion x[n]=y[n]−x[n−1] gives uniqueness only when a boundary value such as x[n0−1] is supplied.

   Narration transcript

   Let's test all six properties on the discrete-time system y of n equals x of n plus x of n minus one. This is a two-sample averager that adds the current input to the previous one. Memory: yes — the output depends on x of n minus one, a past sample. Causality: yes — it uses only the present and one past sample, no future inputs. Linearity: yes — H of a x one plus b x two equals a x one of n plus a x one of n minus one plus b x two of n plus b x two of n minus one, which equals a H of x one plus b H of x two. Time invariance: yes — shifting x gives x of n minus n zero plus x of n minus one minus n zero, which is exactly y of n minus n zero. Stability: yes — if the absolute value of x of n is bounded by M, then the absolute value of y of n is at most two M, which is finite. Invertibility: yes — given y of n and y of n minus one, we can recover x of n recursively. This system is linear and time-invariant, so it is an L T I system.

7. 7. Review

   

   State the input domain and initial conditions before claiming an inverse.

   The two-tap sum is LTI and stable but is not unconditionally invertible on bilateral sequences. For zero-state LTI systems, the impulse response provides the full input/output operator under suitable convolution conditions.

   Narration transcript

   Today we covered the six fundamental properties of systems. Memory: does the output depend on past or future inputs? Causality: does the output depend only on past and present inputs, never future? Linearity: does the system satisfy superposition — homogeneity and additivity? Time invariance: does a time shift in the input produce the same shift in the output? Stability: does every bounded input produce a bounded output? Invertibility: can we uniquely recover the input from the output? A system that is both linear and time-invariant is called an L T I system. L T I systems are central to signals and systems because they can be fully characterized by their impulse response — a topic we'll explore in the next lessons.

Source video: Signals & Systems #18 | Systems & Their Properties - LTI Systems (7:33)