A new proof of the discrete-time bounded-real lemma and lossless bounded-real lemma

Recently, Vaidyanathan [1] presented the discrete-time bounded-real lemma and lossless bounded-real lemma. He derived the lemmas employing energy-balance arguments. The present paper provides an alternative proof of these lemmas. The new proof is very similar to that employed previously [2], [3] in connection with the proof of the discretetime version of the MKY lemma. Note: The notations used in the present paper have the same meaning as in [1].