a property that will become useful in the derivation: where denotes the conjugate transpose of , and denotes expectation. Let us call our output, , the inner product of our filter and the observed signal such that We now define the signal-to-noise ratio, which is our objective function, to be the ratio of the power of the output due to the desired signal to the power of the output due to the noise: We rewrite the above: We wish to maximize this quantity by choosing . Expanding the denominator of our objective function, we have Now, our becomes We will rewrite this expression with some matrix manipulation. The reason for this seemingly counterproductive measure will become evident shortly. Exploiting the Hermitian symmetry of the covariance matrix , we can write We would like to find an upper bound on this expression. To do so, we first recognize a form of the Cauchy–Schwarz inequality: which is to say that the square of the inner product of two vectors can only be as large as the product of the individual inner products of the vectors. This concept returns to the intuition behind the matched filter: this upper bound is achieved when the two vectors and are parallel. We resume our derivation by expressing the upper bound on our in light of the geometric inequality above: Our valiant matrix manipulation has now paid off. We see that the expression for our upper bound can be greatly simplified: We can achieve this upper bound if we choose, where is an arbitrary real number. To verify this, we plug into our expression for the output : Thus, our optimal matched filter is We often choose to normalize the expected value of the power of the filter output due to the noise to unity. That is, we constrain This constraint implies a value of , for which we can solve: yielding giving us our normalized filter, If we care to write the impulse response of the filter for the convolution system, it is simply the complex conjugate time reversal of the input . Though we have derived the matched filter in discrete time, we can extend the concept to continuous-time systems if we replace with the continuous-time autocorrelation function of the noise, assuming a continuous signal , continuous noise , and a continuous filter . Derivation via Lagrangian Alternatively, we may solve for the matched filter