Matched Filters By: Andy Wang
What is a matched filter? (1/1) • A matched filter is a filter used in communications to “match” a particular transit waveform. • It passes all the signal frequency components while suppressing any frequency components where there is only noise and allows to pass the maximum amount of signal power. • The purpose of the matched filter is to maximize the signal to noise ratio at the sampling point of a bit stream and to minimize the probability of undetected errors received from a signal. • To achieve the maximum SNR, we want to allow through all the signal frequency components, but to emphasize more on signal frequency components that are large and so contribute more to improving the overall SNR.
Deriving the matched filter (1/8) • A basic problem that often arises in the study of communication systems is that of detecting a pulse transmitted over a channel that is corrupted by channel noise (i.e. AWGN) • Let us consider a received model, involving a linear time-invariant (LTI) filter of impulse response h(t). • The filter input x(t) consists of a pulse signal g(t) corrupted by additive channel noise w(t) of zero mean and power spectral density No/2. • The resulting output y(t) is composed of go(t) and n(t), the signal and noise components of the input x(t), respectively. Signal g(t) LTI filter of impulse response h(t) x(t) y(t) y(T) ∑ Sample at time t = T Linear receiver White noise w(t)
Deriving the matched filter (2/8) • Goal of the linear receiver • To optimize the design of the filter so as to minimize the effects of noise at the filter output and improve the detection of the pulse signal. • Signal to noise ratio is: where |go(T)|2 is the instantaneous power of the filtered signal, g(t) at point t = T, and σn2 is the variance of the white gaussian zero mean filtered noise.
Deriving the matched filter (3/8) • We sampled at t = T because that gives you the max power of the filtered signal. • Examine go(t): • Fourier transform
Deriving the matched filter (4/8) • Examine σn2: but this is zero mean so and recall that autocorrelation at autocorrelation is inverse Fourier transform of power spectral density
Deriving the matched filter (5/8) filter • Recall: H(f) SX(f) SX(f)|H(f)|2= SY(f) • In this case, SX(f) is PSD of white gaussian noise, • Since Sn(f) is our output:
Deriving the matched filter (6/8) • To maximize, use Schwartz Inequality. Requirements: In this case, they must be finite signals. This equality holds if φ1(x) =k φ2*(x).
Deriving the matched filter (7/8) • We pick φ1(x)=H(f) and φ2(x)=G(f)ej2πfT and want to make the numerator of SNR to be large as possible maximum SNR according to Schwarz inequality
Deriving the matched filter (8/8) • Inverse transform • Assume g(t) is real. This means g(t)=g*(t) • If then for real signal g(t) through duality • Find h(t) (inverse transform of H(f)) h(t) is the time-reversed and delayed version of the input signal g(t). It is “matched” to the input signal.
What is a correlation detector? (1/1) Detector • A practical realization of the optimum receiver is the correlation detector. • The detector part of the receiver consists of a bank of M product-integrators or correlators, with a set of orthonormal basis functions, that operates on the received signal x(t) to produce the observation vector x. • The signal transmission decoder is modeled as a maximum-likelihood decoder that operates on the observation vector x to produce an estimate, . Signal Transmission Decoder
The equivalence of correlation and matched filter receivers (1/3) • We can also use a corresponding set of matched filters to build the detector. • To demonstrate the equivalence of a correlator and a matched filter, consider a LTI filter with impulse response hj(t). • With the received signal x(t) used as the filter output, the resulting filter output, yj(t), is defined by the convolution integral:
The equivalence of correlation and matched filter receivers (2/3) • From the definition of the matched filter, we can incorporate the impulse hj(t) and the input signal φj(t) so that: • Then, the output becomes: • Sampling at t = T, we get:
The equivalence of correlation and matched filter receivers (3/3) Matched filters • So we can see that the detector part of the receiver may be implemented using either matched filters or correlators. The output of each correlator is equivalent to the output of a corresponding matched filter when sampled at t = T. Correlators