Consider a signal s(t) and a lowpass filter with real impulse response h(t), whose FTs are...

 

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Consider a signal s(t) and a lowpass filter with real impulse response h(t), whose FTs are S(v) and H(v). B(t, ƒ) = F¯¹{S(v)H(v − ƒ)} is the complex output of a bandpass filter with center frequency +f. The input signal may be synthesized as the sum of the outputs of such filter banks. The square magnitude of B(t,f) is called the sonograph or sonogram. Prove that the spectrogram of a STFT (for a signal s(t) and window y(t)) and sonograph are equal if the window function of the spectrogram is real and even and equal to the impulse response of the sonograph filter for f=0. Consider a signal s(t) and a lowpass filter with real impulse response h(t), whose FTs are S(v) and H(v). B(t, ƒ) = F¯¹{S(v)H(v − ƒ)} is the complex output of a bandpass filter with center frequency +f. The input signal may be synthesized as the sum of the outputs of such filter banks. The square magnitude of B(t,f) is called the sonograph or sonogram. Prove that the spectrogram of a STFT (for a signal s(t) and window y(t)) and sonograph are equal if the window function of the spectrogram is real and even and equal to the impulse response of the sonograph filter for f=0.