Homework 3 Solutions

Question 1

a) Invertibility

Let $ x_1[n]=0 $ for all $ n $ be an input to the given system. Then, its response is $ y_1[n]=0 $ for all $ n $.

Let $ x_2[n]=\delta [n] $ be an input to the given system. Then, its response is $ y_2[n]=0 $ for all $ n $.

Since $ x_2[n]\neq x_1[n] $ and $ y_2[n]=y_1[n] $, then the system is not invertible.

Memory:

The output $ y[n] $ depends on past values of $ x[n] $, since we have $ x[n-1] $ in the system equation.

Hence, we deduce that system has memory.

Causality:

The output $ y[n] $ depends only on the current ( $ x[n] $ ) and past ( $ x[n-1] $ ) values of the input.

Hence, the given system is causal.

Stability

Let $ x[n] $ be a bounded signal by some number B, i.e. $ |x[n]|<B $ for all $ n $.

Then the response to $ x[n] $ is always bounded as such: $ |y[n]|<B^2 $ for all $ n $.

Thus the given system is stable.

Linearity

Let $ x_1[n] $ be an input to the given system. Then its response is $ y_1[n]=x_1[n]x_1[n-1] $.

Now, let $ x_2[n]=ax_1[n] $ be an input to the system, where $ a $ can be any number. Then its response is $ y_2[n]=a^2x_1[n]x_1[n-1]\neq ay_1[n] $, then the system is not linear.

Time invariance

Let $ x_1[n] $ be an input to the system. Then $ y_1[n]=x_1[n]x_1[n-1] $ is its response.

Now, let $ x_2[n]=x_1[n-n_0] $ be an input to the system, where $ n_0 $ can be any number. Then, $ y_2[n]=x_1[n-n_0]x_1[n-1-n_0]=y_1[n-n_0] $.

Hence the system is time invariant.

b) Invertibility

Let $ x_1(t)=0 $ for all $ t $ be an input to the given system. Then, its response is $ y_1(t)=0 $ for all $ t $.

Let $ x_2(t)=\delta (t-2) $ be an input to the given system. Then, its response is $ y_2(t)=0 $ for all $ t $ since $ -1\leq\sin(t)\leq 1 $.

Since $ x_2(t)\neq x_1(t) $ and $ y_2(t)=y_1(t) $, then the system is not invertible.

Memory:

For $ t=-\pi/2 $, we have $ y(-\pi/2)=x(-1) $. However, $ -\pi/2\leq -1 $.

Then the output $ y(t) $ depends on future values of the input $ x(t) $.

Hence, we deduce that system has memory.

Causality:

Using the same example for the memory part, we can say that the system is non-causal.

Stability

Let $ x(t) $ be a bounded signal by some number B, i.e. $ |x(t)|<B $ for all $ t $.

Then the response to $ x(t) $ is obviously always bounded as such: $ |y(t)|<B $ for all $ t $.

Thus the given system is stable.

Linearity

Let $ x_1(t) $ be an input to the given system. Then its response is $ y_1(t)=x_1(\sin(t)) $.

Let $ x_2(t) $ be an input to the given system. Then its response is $ y_2(t)=x_2(\sin(t)) $.

Now, let $ x_3(t)=ax_1(t)+bx_2(t) $ be an input to the system, where $ a $ and $ b $ can be any number. Then its response is $ y_2(t)=ax_1(\sin(t))+bx_2(\sin(t))=ay_1(t)+by_2(t) $.

Hence the system is linear.

Time invariance

Let $ x_1(t) $ be an input to the system. Then $ y_1(t)=x_1(\sin(t)) $ is its response.

Now, let $ x_2(t)=x_1(t-t_0) $ be an input to the system, where $ t_0 $ can be any number. Then, $ y_2(t)=x_1(\sin(t-t_0))=y_1(t-t_0) $.

Hence the system is time invariant.