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DTFT and Convolution

Module by: Richard Baraniuk

Summary: An overview of the DTFT and Convolution.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Figure 1
Figure 1 (fig1.png)

Given:

Xⅇⅈω X ω and Hⅇⅈω H ω , compute Yⅇⅈω Y ω
Yⅇⅈω=∑n=-∞∞ynⅇ-ⅈωn=∑n=-∞∞∑k=-∞∞xkhn−kⅇ-ⅈωn=∑k=-∞∞xk∑n=-∞∞hn−kⅇ-ⅈωn 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 (1)
Let m=n−k→n=m+k m n k n m k
=∑k=-∞∞xk∑m=-∞∞hmⅇ-ⅈωm+k=∑k=-∞∞xkⅇ-ⅈωk∑m=-∞∞hmⅇ-ⅈωm k x k m h m ω m k k x k ω k m h m ω m (2)
Yⅇⅈω=XⅇⅈωHⅇⅈω Y ω X ω H ω (3)
Figure 2
Figure 2 (fig2.png)
x*h ↔ DTFT XH x h ↔ DTFT X H (4)

Example 1: Lowpass Filtering of an ECG Signal to Suppress Noise

Figure 3
Figure 3 (fig3.png)
ecg
  • voltages are very small ( μV 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
Figure 4
Figure 4 (fig4.png)
and corrupts it with additive NOISE
Figure 5
Figure 5 (fig5.png)
so that the actual received signal is SIGNAL+NOISE
Figure 6
Figure 6 (fig6.png)
This is the signal we actually measure. It is "noisy".

Key Point:

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

Figure 7
Figure 7 (fig7.png)

Averaging Filter h Physical Interpretation

h[n] h[n] is a moving average

  • smooths the data
  • h[n] h[n] is a LOWPASS FILTER
We see this behavior in Hω H ω . Also as 2M+1 2 M 1 increases, the cutoff frequency ω 1 =π2M+1 ω 1 2 M 1 decreases which leads to more smoothing.
Figure 8
Figure 8 (fig8.png)
LOWPASS FILTERING = SMOOTHING = AVERAGING

Result

Figure 9
Figure 9 (fig9.png)
Bottom Line: Figure 9 looks much better than Figure 6

At home:

  1. Interpret the filtering action in terms of hn h n (time/sample domain) and Hω H ω (frequency domain)
  2. What price did we pay?
Figure 10
SUMMARY
SUMMARY (fig10.png)

Extension 1:

Can we improve on the frequency response of the moving average filter?

Extension 2:

ECG also picks up plenty of 60Hz interference

Figure 11
Figure 11 (fig12.png)
How would you combat this interference using an LTI filter?

Glossary

electrocardiogram:
measure electrical signal given off by the heart.

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