DTFT and Convolution 1.2 2004/06/18 09:25:49 GMT-5 2004/07/20 10:33:26 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Robert Ahlfinger ahlfing@rice.edu Convolution DTFT ELEC 301 An overview of the DTFT and Convolution.

X ω and H ω , compute Y ω Y ω n y n ω n n k x k h n k ω n k x k n h n k ω n Let m n k n m k k x k m h m ω m k k x k ω k m h m ω m Y ω X ω H ω

x h ↔ DTFT X H Lowpass Filtering of an ECG Signal to Suppress Noise

ecg voltages are very small ( μ V !) therefore there is plenty of measurement noise use DSP to reduce noise simple processing: low pass filtering Measurement process takes the TRUE SIGNAL

and corrupts it with additive NOISE

so that the actual received signal is SIGNAL+NOISE

This is the signal we actually measure. It is "noisy". Signal is in low frequencies and noise is spread over all frequencies We need a lowpass filter to suppress noise. Try a simple averaging filter

Averaging Filter h Physical Interpretation h[n] is a moving average smooths the data h[n] is a LOWPASS FILTER We see this behavior in H ω . Also as 2 M 1 increases, the cutoff frequency ω 1 2 M 1 decreases which leads to more smoothing.

LOWPASS FILTERING = SMOOTHING = AVERAGING

Result

Bottom Line: looks much better than At home: Interpret the filtering action in terms of h n (time/sample domain) and H ω (frequency domain) What price did we pay?

SUMMARY

Extension 1: Can we improve on the frequency response of the moving average filter? Extension 2: ECG also picks up plenty of 60Hz interference

How would you combat this interference using an LTI filter? electrocardiogram measure electrical signal given off by the heart.