Sinusoidal “Dot-Products”

• The operation described in the previous section is referred to as a “dot product” or “inner product”. More generally, the inner product of two real-valued discrete-time signals $x[n]$ and $y[n]$ is given by
    $$\langle x_n, y_n \rangle = \sum_{n=0}^{N-1} x[n] y[n].$$ (3)

• As noted above, $\langle x_n, x_n \rangle = N/2$ when $x[n]$ is a sinusoidal signal of exactly one period of length $N$ and an amplitude equal to one.

• Figure 3 shows the sinusoidal signal from Fig. 1 and another signal of frequency 4 times the first.

  Figure 3: Two discrete-time sinusoidal signals.

  

• Importantly, the inner product of the signals of Fig. 3 is zero.

• This result can be generalized to all cases for discrete-time sinusoidal signals where: 1. Both period lengths are integer values; 2. Both signals contain an exact integer number of periods; and 3. The two signals are of different period lengths (or equivalently, of different frequencies).

• This result is interesting because it suggests that, given a signal composed of many sinusoidal components, we can perform a sequence of dot-products with sinusoids of different frequencies to determine which are present in the signal. That is, non-zero inner product results will indicate the presence of a particular frequency in the signal.

• In addition, the inner product is proportional to the amplitude of the signals. For example, $\langle x_n, 0.5x_n \rangle = N/4$ when $x[n]$ is a sinusoidal signal of exactly one period of length $N$.

• Thus, we can “compare” a length $N$ signal to a set of discrete-time sinusoidal components given by the normalized radian frequencies of $\hat{\omega}_k = \frac{2\pi}{N} k$ for $k = 0, 1, \ldots, N-1$ (the actual frequencies are given by $\omega_k = \hat{\omega}_k f_s$, where $f_s$ is the sample rate).

• Note that these sinusoidal components are all integer multiples of a “fundamental” given by $\hat{\omega}_1 = \frac{2\pi}{N}$.

• The $\hat{\omega}_0$ component allows us to compute the average, or DC, value of a signal (if we divide the inner product result by $N$).

• Before we get too excited, we should note what can happen to the dot-product when the phase of the two signals is not equal.

• In particular, the dot-product of our original signal $w[n]$ and a phase shifted-version of itself produces a result that depends on the amount of phase shift, as shown in Table 1.
  

  Table 1: Inner product values for $w[n]$ and $w[n]$ phase-shifted by $\phi$ radians.

  Phase Shift $\phi$ (radians)	$\langle w_{n}, w_{n+\phi} \rangle$ ($N=32$)
  0	16
  $\pi / 4$	11.3
  $\pi/2$	0
  $3\pi / 4$	-11.3
  $\pi$	-16
  $5\pi / 4$	-11.3
  $3\pi/2$	0
  $7\pi / 4$	11.3

  
  
  
  

• Thus, a sinusoidal component could go completely undetected when using the dot-product technique suggested above if it is phase-shifted a quarter cycle relative to the “comparison” sinusoid of the same frequency.

• On the other hand, we are furtunate that the dot-product of sinusoids of different frequencies is not dependent on phase shifts. That is, the result is always zero when the two sinusoidal signal frequencies are unequal, no matter their relative phases.