Let x 1 (t) and x 2 (t) be two real valued signals Show that the square of the norm of the signal x 1 (t) x 2 (t) is the sum of the square of the norm of x 1 (t) and the square of the norm of x 2 (t), if and only if x 1 (t) and x 2 (t) are orthogonal that is, x 1 x 2 2 x 1 2 x 2 2 if and only if (x 1 , x 2 ) 0 Note the analogy to vectors in three dimensional space the Pythagorean theorem applies only to vectors that are orthogonal or perpendicular (zero dot product)

Question: Let x 1 (t) and x 2 (t) be two real-valued signals. Show that the square of the norm of the signal x 1 (t)

Let x₁(t) and x₂(t) be two real-valued signals. Show that the square of the norm of the signal x₁(t) + x₂(t) is the sum of the square of the norm of x₁(t) and the square of the norm of x₂(t), if and only if x₁(t) and x₂(t) are orthogonal; that is, ||x₁ + x₂||² = ||x₁||² + ||x₂||² if and only if (x₁, x₂) = 0. Note the analogy to vectors in three dimensional space: the Pythagorean theorem applies only to vectors that are orthogonal or perpendicular (zero dot product).

Step by Step Solution

★★★★★

3.25 Rating (151 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Since the signals are assumed real From thi... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Principles Communications Systems Questions!

Consider a pair of complex-valued signals s 1 (t) and s 2 (t) that are respectively represented by s 1 (t) = a 11 ? 1 (t) + a 12 ? 2? (t), ? ? 2 (t) = a 21 ? 1 (t) + a 22 ? 2? (t), ? ? 1 (t) and ? 2...

Figure a, shows a non-coherent receiver using a matched filter for the detection of a sinusoidal signal of known frequency but random phase, in the presence o additive white Gaussian noise. An...

Two finite-length signals, x 1 [n] and x 2 [n], are sketched in Figure. Assume that x 1 [n] and x 2 [n] are zero outside of the region shown in the figure. Let x 3 [n] be the eight-point circular...

1: GPS AMS 103, Spring 2017 Due: Thursday, February 9, 11:59 pm EST Quiz: Tuesday, February 14, in class Objectives To reinforce concepts relevant to GPS technology and the underlying mathematical...

The 3-point Moving Average Filter given by 1 y[n] = [x[n] + x[n 1] + x[n 2]] (1) can be used to clean up' a signal that has been corrupted by a noise signal. To measure the effectiveness of the...

Let A, B be sets. Define: (a) the Cartesian product (A B) (b) the set of relations R between A and B (c) the identity relation A on the set A [3 marks] Suppose S, T are relations between A and B, and...

The Basics of Financial Mathematics Spring 2003 Richard F. Bass Department of Mathematics University of Connecticut c These notes are 2003 by Richard Bass. They may be used for personal use or class...

Math\t107-6381\t-\tQuiz\t#4\t-\tSchultz\t-\tDue\tFebruary\t21,\t2016\t-\tpage\t1\tof 3 Follow\tthese\tdirections\tcarefully. This\tquiz\tis\tdue\tby\t11:59\tEastern\ttime\ton\tFebruary\t21,\t2016. o...

Consider two real-valued signals si(t) and s2(t) whose pre-envelopes are denoted by S1+(t) and s2+(1), respectively. (a) Please show that ** Re[s4 (1)]Re[$+ (1)]dt ==Re [S (1)]s (1)dt ] 2+ (b)...

Each question is worth 5 points. To receive full credit for a correct answer please, show your work and processes in solving each question. Total points = 125. 1- Simply the expression: (5X3); x y 2-...

A component to be used in the assembly of a transmission mechanism is manufactured in a process for which the two quality characteristics of tensile strength (X1) and diameter (X2) are of importance....

Using book value to measure profitability and to value a company's stock has limitations. Discuss five such limitations from an accounting perspective. Be specific.

Assume that Netflix s cost of revenues consists only of variable costs, and all other operating expenses ( marketing , technology and development, and general and administrative ) are fixed. Further,...

Exercise 21.4 BUSINESS COMBINATION VALUATION AND PRE-ACQUISITION ENTRIES On 1 July 2013, Pyxis Ltd acquired all the share capital of Gemini Ltd for 218 500. At this date, Gemini Ltds equity...

When radiation having a Stokes vector S = [S 0 , S 1 , S 2 , S 3 ] is incident on an antenna that receives only linearly x-polarised radiation, the detected power is proportional to S.P, where P =...

Calculate the ratio of the spectral radiances of black bodies at 300 K and 6000 K at (i) A wavelength of 0.1 m, (ii) A wavelength of 1 m, (iii) A frequency of 1000 GHz, (iv) A frequency of 1 GHz.

Show that the Fourier transform of the Gaussian function is proportional to and interpret the result. - ) f(1) = exp 202 ioto o?o exp 2

Horton Housewares' accounts receivable decreased during the year by $13.0 million. Required: What is the amount of cash Horton received from customers during the reporting period if its sales were...

It is acceptable to record prepayment of expenses as debits to expense accounts, if the unused or unexpired portions at the end of an accounting period are transferred to prepaid expense (asset)...

1 . There are five vital statistics derived from raw population data that indicate the overall health of a country.List and define the five 'vital statistics' .Describe how each of the five are...