Size of A Signal: Norms 1.2 2004/08/02 09:28:19 GMT-5 2006/09/23 16:44:53.308 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Robert Ahlfinger ahlfing@rice.edu Elec 301 Norm A module concerning the size of a signal, more specifically norms. "Size" indicates largeness or strength. We will use the mathematical concept of the norm to quantify this notion for both continuous-time and discrete-time signals. First we consider a way to quantify the size of a signal which may already be familiar.

Continuous-Time Energy Our usual notion of the energy of a signal is the area under the curve f t 2

E f t f t 2 Calculate E f for

Generalized Energy: Norms The notion of "energy" can be generalized through the introduction of the L p norm: p f t f t p 1 p where 1 p . E f 2 f 2 Calculate the L p norm of

What happens to p f t f t p 1 p as p ?

L ∞ norm ∞ f ess sup f t

Discrete-Time Energy

E f n f n 2 p f t f t p 1 p where 1 p f n f n p f p n 0 N 1 f n p where 1 p f n 0, 1, ..., N-1 f n

Finite Norm Signals What are the conditions on a signal for p f ? Look at all 4 fundamental signal classes

Discrete-Time and Finite Length

This is a length N vector. f f 0 f 1 f 2 f ... f N 1 f f 0 f f 1 f f 2 ... f f N 1 where f N , or f N N -dimensional complex or real Euclidean space. N 3 , f is a real signal.

l p 0 N 1 f N p f but from previous discussion l p 0 N 1 N

Discrete-Time and Infinite Length

can still interpret f as an infinite-length vector f f ... f -1 f 0 f 1 f 2 f ... but , R don't make sense. l p z f p f p f p n f n p where 1 p f n z f n What does it take for an f to be in l p z ? Sketch an f l p z and f l p z . What characteristics does f l p z have and what happens as we charge p ?

Continuous-Time and Finite-Length

We will still refer to f t as a vector; more on this later. L p T 1 T 2 f T 1 T 2 p f p f t T 2 T 1 f t p 1 p where 1 p p f ess sup f t where T 1 t T 2 What does it take for and f to be in L p T 1 T 2 ?

Continuous-Time and Infinite-Length

We will still refer to f t as a vector. L p R f p f p f t f n p 1 p where 1 p f ess sup f t where t What does it take for an f L p R ? Sketch an f L p R and f L p R .

Power What do we do when p f ? Periodic Signal

Solution: Look at the "norm per unit time". ie norm over one period. ie norm of infinite-length signal converted to finite length signal by windowing.

p f T is the measure. Units for p 2 ? L 2 Power "energy per unit time" Useful when E f time average of energy P f T T t T 2 T 2 f t 2

compute Energy T 2 f 2 T look at T Energy T 2 f 2 T P f is often called the mean-square value of f . P f is called the root mean squared (RMS) value of f . Units? "Energy signals" have finite norm (energy) E f . "Power signals" have finite and nonzero power P f , P f 0 , and ( E f ).

Conclusions Energy signals are not power signals. Power signals are not energy signals. Why? Are all signals either energy or power signals? f t t

The 4 fundamental classes of signals we will study depend on the independent (time) variable.